Inserting Digits

28269 is a composite number with the interesting property that if we insert the digit 7 in any position, the result is a prime number. Thus 728269, 278269, 287269, 282769, 282679 and 282697 are all prime. Such numbers belong to OEIS A216168 (permalink):

A216168: composite numbers and 1 which yield a prime whenever a 7 is inserted anywhere in them, including at the beginning or end.

Up to 40000, the members of the sequence are:

1, 9, 27, 33, 39, 57, 87, 159, 177, 187, 603, 717, 753, 949, 1257, 1707, 2277, 2367, 4317, 4623, 4779, 4797, 5773, 6757, 6777, 7017, 7471, 7479, 7747, 7797, 7813, 7977, 8797, 9777, 9987, 10777, 11757, 17679, 28269, 28437, 29779, 34177, 34771

There is an associated OEIS sequence for inserting the digit 1. It is OEIS A216165 (permalink):

A216165: composite numbers and 1 which yield a prime whenever a 1 is inserted anywhere in them, including at the beginning or end.

The initial members of this sequence are:

1, 49, 63, 81, 91, 99, 117, 123, 213, 231, 279, 319, 427, 459, 621, 697, 721, 801, 951, 987, 1113, 1131, 1261, 1821, 1939, 2101, 2149, 2211, 2517, 2611, 3151, 3219, 4011, 4411, 4887, 5031, 5361, 6231, 6487, 7011, 7209, 8671, 9141, 9801, 10051, 10161, 10281, 10603, 10921, 11121, 11127, 11211, 11641, 11767, 11791, 11869, 12997, 13111, 13143, 14311, 16911, 17023, 17541, 18081, 18619, 19677, 21039, 21711, 23289, 25197, 29169, 29971, 31111, 34777, 38559

Here is the associated OEIS A216166 sequence for inserting a 3 (permalink).

A216166: composite numbers and 1 which yield a prime whenever a 3 is inserted anywhere in them (including at the beginning or end).

The initial members of this sequence are:

1, 121, 343, 361, 533, 637, 793, 889, 943, 1183, 3013, 3223, 3353, 3403, 3757, 3827, 3893, 4313, 4543, 4963, 8653, 10423, 14257, 20339, 23083, 23419, 30917, 33031, 33101, 33323, 33433, 33701, 33821, 34333, 34393, 35453, 36437, 36533, 39137, 39247

Here is the result for inserting the digit 9. I wasn't able to locate the OEIS sequence (permalink).

91, 209, 539, 749, 923, 931, 1079, 1139, 2717, 2959, 3971, 3979, 4559, 5629, 6401, 6739, 8213, 8491, 8939, 9607, 11089, 11227, 13943, 14269, 14371, 17381, 17689, 24059, 25517, 25937, 25949, 29087, 29197, 29419, 30989, 31691

There are various variations on this theme. Once can look at numbers, both composite and prime, or one can look at only primes. The insertion can be between digits but not at the beginning and end. The insertion can be only at the beginning and end etc. Let's consider one of these: OEIS A216167.

A216167: composite numbers which yield a prime whenever a 5 is inserted anywhere in them, excluding at the end.

Here are the initial members of the sequence:

9, 21, 57, 63, 69, 77, 87, 93, 153, 231, 381, 407, 413, 417, 501, 531, 581, 651, 669, 741, 749, 783, 791, 987, 1241, 1551, 1797, 1971, 2189, 2981, 3381, 3419, 3591, 3951, 4083, 4503, 4833, 4949, 4959, 5049, 5117, 5201, 5229, 5243, 5529, 5547, 5603, 5691, 5697, 50 6957, 7329, 7389, 7557, 8451, 8711, 9561, 9617, 11337, 11631, 13511, 13533, 15153, 17991, 19539, 23553, 25869, 27053, 30093, 31551, 32249, 32951, 36441, 38159

This approach could be extended to biprimes. For example, one could ask what biprimes (or semiprimes) remain biprimes when the digit 1 is inserted anywhere in the number, including its beginning and end? Using Gemini, here are results that were generated:

Biprimes up to 40000 maintaining the property upon digit insertion (permalink):

Digit 0:

None found in this range.

Digit 1:

[34, 55, 77, 85, 87, 111, 119, 121, 141, 219, 415, 417, 514, 537, 591, 689, 713, 717, 718, 731, 781, 835, 841, 921, 1111, 1114, 1138, 1227, 1293, 1357, 1391, 1591, 1643, 1671, 1761, 1851, 1915, 2171, 2181, 2533, 2623, 2631, 3207, 3561, 3579, 3811, 4119, 4171, 4267, 4353, 5223, 5311, 5321, 5353, 6157, 6181, 6429, 6621, 6697, 7087, 7113, 7131, 7251, 7311, 7513, 7531, 7571, 7897, 8201, 8254, 8511, 8659, 9111, 9123, 9167, 9247, 9271, 9487, 9651, 10147, 10217, 10401, 10417, 10699, 10743, 10765, 11013, 11101, 11107, 11391, 11479, 11513, 11787, 11819, 11899, 12219, 12381, 12399, 12709, 12718, 12751, 12773, 14109, 14119, 14527, 14711, 14727, 15261, 15711, 16213, 16251, 16489, 16621, 16751, 16801, 16917, 17151, 17461, 17527, 17601, 17677, 18091, 18519, 18591, 19111, 19117, 19711, 19729, 19741, 19858, 19959, 20311, 21103, 21461, 21829, 22601, 23811, 24159, 24161, 25331, 26283, 26521, 26869, 27087, 27831, 28009, 29487, 30061, 30157, 30499, 31273, 31587, 32521, 32599, 33193, 33953, 34051, 34567, 34663, 35941, 36331, 36631, 37081, 37461, 37891, 38067, 38623, 39478, 39793]

Digit 2:

[6, 26, 62, 206, 302, 1202, 1226, 1262, 2966, 12242, 26762, 32282, 36422]

Digit 3:

[9, 14, 39, 55, 58, 93, 94, 133, 235, 274, 291, 305, 309, 314, 365, 403, 453, 554, 649, 713, 763, 1067, 1133, 1157, 1385, 1589, 1631, 1961, 1969, 2073, 2174, 2231, 2419, 2483, 2723, 2811, 2867, 3035, 3043, 3134, 3239, 3351, 3443, 3455, 3518, 3595, 3974, 3991, 4315, 4387, 4478, 5057, 5249, 5545, 5761, 6071, 6233, 6431, 6731, 6807, 6817, 6893, 6953, 7033, 7067, 7403, 7543, 7903, 8141, 8359, 8438, 8473, 8483, 8495, 8683, 9395, 9713, 9731, 9838, 9943, 10609, 10897, 11107, 11179, 13483, 13531, 13953, 14131, 14473, 14515, 14809, 15163, 15397, 15833, 16049, 16331, 17113, 17281, 17311, 17521, 17767, 18613, 18721, 19331, 19651, 19693, 20191, 20831, 21541, 22121, 22163, 22733, 23107, 23237, 23303, 23503, 23533, 24173, 24283, 24797, 25129, 25843, 26173, 26545, 27389, 27833, 28399, 29903, 30353, 30445, 30779, 30973, 31187, 31309, 31463, 31757, 31861, 32023, 32231, 32293, 32483, 32539, 32723, 32861, 33017, 33134, 33163, 33193, 33238, 33307, 33313, 33314, 33373, 33407, 33443, 33643, 33689, 33833, 33881, 33951, 33953, 33977, 33979, 34003, 34201, 34249, 34633, 34837, 34885, 35063, 35303, 35723, 35887, 36283, 36289, 36893, 37033, 37333, 37351, 37391, 37733, 38057, 38939, 39449, 39497, 39513, 39793]

Digit 4:

[9, 69, 93, 115, 319, 321, 381, 427, 471, 489, 511, 529, 535, 559, 1101, 1383, 1441, 1623, 1909, 2173, 2319, 2361, 2761, 2881, 3409, 3817, 4009, 4141, 4413, 4479, 4749, 5029, 5143, 5299, 5455, 5611, 6249, 6313, 6331, 6423, 6459, 6541, 7063, 7423, 8097, 8403, 8797, 9313, 11029, 11787, 11881, 12063, 13051, 13389, 13449, 13641, 14187, 14473, 14493, 15529, 18247, 21427, 23611, 24643, 25351, 26409, 26457, 26989, 27157, 28761, 30531, 31609, 31987, 34627, 35493, 35713, 36019]

Digit 5:

[15, 35, 51, 65, 155, 219, 299, 411, 515, 545, 554, 713, 755, 771, 818, 893, 905, 965, 993, 1055, 1469, 2651, 3005, 3065, 3953, 4313, 4359, 4811, 5033, 5069, 5123, 5129, 5345, 5429, 5513, 5543, 5585, 5891, 6218, 7053, 8051, 8301, 8553, 8945, 9155, 9543, 10109, 10749, 16535, 16595, 16955, 17529, 17555, 18653, 19451, 21449, 23255, 24263, 25061, 25131, 25293, 25751, 27831, 28529, 30993, 31539, 33551, 33933, 34559, 35411, 35681, 35693, 35921]

Digit 6:

[22, 62, 122, 662, 746, 2066, 17522, 38062]

Digit 7:

[377, 471, 721, 737, 778, 799, 849, 961, 1057, 1267, 1273, 1639, 1651, 1717, 2041, 2257, 2458, 2841, 3099, 3763, 3977, 3981, 4083, 4247, 4281, 4467, 4577, 4757, 5497, 6009, 6087, 6127, 6267, 6297, 6341, 6729, 6787, 7117, 7313, 7739, 7921, 7971, 8063, 8097, 8457, 8749, 8907, 9057, 9313, 9469, 9897, 9899, 10381, 10617, 11013, 11107, 11217, 11283, 12273, 13017, 13749, 14277, 14757, 15529, 16147, 16957, 17071, 17179, 17439, 17503, 17517, 17637, 17701, 17773, 17781, 17809, 18283, 19039, 19857, 21477, 21741, 23731, 24537, 27147, 27433, 27769, 28263, 28999, 29163, 29229, 30007, 30127, 30721, 31753, 31879, 32259, 32937, 33163, 33897, 34467, 35881, 37381, 37803, 37837]

Digit 8:

[15, 69, 95, 119, 213, 341, 843, 1337, 2481, 2831, 3057, 3161, 3489, 3513, 3587, 3849, 4803, 5489, 5663, 6189, 6459, 6893, 7113, 7355, 7379, 7409, 8057, 8141, 8187, 8189, 8279, 8331, 8399, 8567, 8751, 8889, 8981, 9881, 10119, 11009, 11381, 11477, 12171, 12531, 14109, 16721, 17517, 18087, 18227, 18663, 18809, 20669, 20783, 22467, 23597, 26081, 28299, 28511, 28739, 28781, 28887, 29321, 29891, 30209, 32393, 33881, 34409, 36321, 37581, 38217]

Digit 9:

[4, 21, 33, 51, 74, 93, 95, 115, 119, 183, 235, 247, 259, 481, 589, 799, 813, 914, 917, 959, 979, 989, 993, 995, 998, 1067, 1101, 1165, 1174, 1589, 1799, 1829, 1849, 2098, 2249, 2257, 2509, 2654, 2839, 2923, 2977, 3029, 3099, 3421, 3781, 3959, 4097, 4319, 4699, 4859, 4897, 5065, 5137, 5293, 5909, 5993, 6049, 6107, 6898, 7837, 7921, 8314, 8359, 8489, 8593, 8921, 9094, 9299, 9329, 9335, 9353, 9509, 9569, 9599, 9691, 9763, 9865, 9959, 9977, 10189, 10489, 10511, 10999, 12139, 12199, 13989, 14311, 14599, 14761, 15011, 15409, 15449, 15829, 16013, 16357, 16969, 17131, 17149, 17177, 17663, 18293, 18383, 18991, 19099, 19187, 19307, 19369, 19493, 19579, 19658, 19667, 19879, 19907, 19959, 20093, 20131, 20765, 21469, 22387, 22597, 22927, 22991, 23299, 24839, 24901, 25003, 25401, 25807, 25927, 26291, 26447, 27667, 27723, 27859, 27877, 28877, 28939, 28981, 28997, 29227, 29321, 29878, 30679, 30767, 30799, 30997, 31909, 32961, 34309, 35215, 35269, 35833, 35939, 36289, 36989, 37249, 38093, 38159, 38641, 39289, 39797]

Let's test for 28299 which is listed under the digit 8 where we find that:

• \(28299 = 3 \times 9433\)
• \(828299 = 23 \times 36013\)
• \(288299 = 11 \times 26209\)
• \(282899 = 79 \times 3581\)
• \(282989 = 7 \times 40427\)
• \(282998 = 2 \times 141499\)

Here the results for sphenic numbers (permalink):

Digit 0:

None found in this range.

Digit 1:

[1595, 3245, 5258, 5710, 7015, 7718, 8255, 8395, 9185, 9878, 9915, 11018, 11194, 16511, 17005, 18518, 18758, 18778, 19245, 27911, 30354, 31665, 33115, 33514, 37614, 38129, 39179]

Digit 2:

[222, 282, 1742, 2222, 2586, 2782, 2922, 3286, 3342, 3926, 4342, 4362, 4722, 5066, 5246, 5262, 7582, 7622, 7782, 8706, 9726, 10622, 10642, 11822, 13782, 14126, 15026, 15422, 16402, 16682, 17122, 20802, 21566, 21962, 22026, 22202, 22782, 23026, 25782, 26942, 27122, 27282, 27726, 29926, 30522, 31002, 31922, 32426, 33242, 34302, 34922, 36182, 37022, 38222, 39422, 39482]

Digit 3:

[615, 1245, 1533, 2198, 2289, 2739, 3333, 3355, 3358, 3590, 5034, 5558, 5734, 6730, 7761, 8238, 8931, 9254, 12189, 12207, 12215, 12595, 13334, 15035, 15897, 15933, 16833, 17733, 18381, 18578, 18579, 18867, 20194, 20894, 23834, 24447, 25534, 28634, 28821, 30385, 30747, 30981, 31190, 32349, 32354, 32774, 33303, 33321, 33835, 33843, 33998, 35734, 37205, 37505, 37558, 38001, 38165, 38973, 39058, 39819, 39934]

Digit 4:

[345, 665, 1547, 1955, 2895, 4585, 4695, 5595, 6645, 7545, 7885, 8729, 9465, 9485, 9515, 9685, 9911, 9915, 10885, 12265, 14465, 19245, 19495, 22555, 25991, 26381, 26745, 29087, 32035, 32145, 32155, 35405]

Digit 5:

[385, 555, 595, 705, 1085, 1455, 2085, 2355, 2555, 2685, 2865, 3355, 3558, 3565, 3585, 4345, 4355, 4565, 4605, 5258, 5405, 5495, 5555, 5835, 5898, 6135, 6785, 6955, 6978, 7554, 7955, 8355, 8533, 9678, 10585, 11473, 12174, 12395, 13115, 13574, 14795, 15535, 17515, 17895, 19598, 20855, 20881, 21765, 21954, 23165, 25085, 25334, 25543, 25858, 26114, 26115, 26555, 26765, 28145, 28754, 29235, 30594, 30635, 30965, 32415, 33155, 33485, 34655, 35315, 35655, 35893, 36515, 36555, 37465, 37565, 37655, 38185, 38558, 39589, 39785, 39885, 39895]

Digit 6:

[186, 266, 582, 806, 906, 1266, 2266, 3562, 3606, 4462, 4966, 5486, 5622, 6086, 6386, 6886, 7966, 9266, 10366, 10586, 13166, 14162, 14806, 16702, 17366, 21626, 22526, 23246, 23446, 26966, 27062, 28262, 31666, 32662, 34226, 34622, 37226, 39742]

Digit 7:

[777, 4330, 4697, 4938, 5678, 5757, 7287, 7777, 7798, 8378, 8898, 9258, 17798, 19877, 23009, 25277, 29530, 29798, 30797, 32514, 36707, 36743, 37077]

Digit 8:

[2185, 2795, 2821, 6335, 8195, 8215, 8255, 8355, 8905, 9285, 10165, 11615, 14885, 18241, 18885, 26315, 30965, 31027, 34177]

Digit 9:

[399, 915, 1515, 2595, 2877, 2937, 2955, 2985, 3201, 3399, 3598, 3945, 4767, 5019, 5133, 5709, 6141, 6693, 7257, 7918, 8697, 8798, 9291, 9398, 9545, 9717, 10090, 10923, 14755, 15171, 15794, 18699, 19495, 19833, 19839, 19869, 19970, 20694, 21754, 21957, 22258, 22989, 23691, 24477, 25203, 25809, 27219, 29289, 29578, 30018, 32165, 34869, 35457, 36879, 39158, 39185]

Let's test for 28262 which is listed under the digit 6:

• \(28262 = 2 \times 13 \times 1087 \)
• \(628262 = 2 \times 53 \times 5927 \)
• \(268262 = 2 \times 113 \times 1187 \)
• \(286262 = 2 \times 41 \times3491 \)
• \(282662 = 2 \times 79 \times 1789 \)
• \(282626 = 2 \times251 \times 563 \)

Super Sphenic Numbers

The number associated with my diurnal age today (28262) is what might be termed a "super sphenic number" as I'll explain in this post. Firstly however, its factorisation:$$28262=2 \times 13 \times 1087$$If we reverse its digits, we get the number 26282 and this number is also sphenic:$$26282 = 2 \times 17 \times 773$$Let's now concatenate the factors of 28262 in ascending order. This gives us the number 2131087 which is also sphenic:$$2131087 = 7 \times 167 \times 1823$$28262 has a sum of digits of 20 and if we add this to the original number we get the palindromic number 28282 which is also sphenic:$$28262+20=28282 = 2 \times 79 \times 179$$The number has a product of digits of 384 and if we subtract this from the original number we get 27878 which is sphenic:$$28262 - 384 = 27878 = 2 \times 53 \times 263$$If we consider only the internal digits of 28262, we get the number 826 which is also sphenic:$$2\, 826 \, 2 \rightarrow826=2 \times 7 \times 59$$When a sphenic number is considered as a sphenic brick then it has an associated number in the form of the brick's surface area. In the case of 28262, this associated surface area is 32662 square units and this number too is sphenic:$$32662 = 2 \times 7 \times 2333$$28262 has a sum of proper divisors that is also sphenic:$$ \sigma(28262) - 28262 =17434 = 2 \times 23 \times 379 $$The number has a sum of prime factors (1102) that is sphenic:$$2 +13+1087=1102 = 2 \times 19 \times 29$$28262 has a totient of 13032 which is not sphenic but its cototient (number - totient) of 15230 is:$$28262 - 13032 = 15230 = 2 \times 5 \times 1523$$So we can see that 28262 may well be termed a super sphenic number because of the above associations.

Semiprimes Within Semiprimes

I got to thinking about semiprimes and how many of them have the property that the digits of their two prime factors can both be rearranged to form two new semiprimes. It's as if two additional semiprimes are hiding within the factors of the original semiprime.

It's simple enough to investigate and I could have written the code but I lazily got Gemini to do it for me. It turns out that there are 1077 such numbers in the range up to 40000 (permalink). the first such number is 361:$$ \begin{align} 361 &=19 \times 19 \\19 &\rightarrow 91 =7 \times 13 \end{align}$$The next such number is not square and is 779:$$ \begin{align} 779 &= 19 \times 41 \\ 19 &\rightarrow 91 =7 \times 13 \\ 41 &\rightarrow 14 =2 \times 7 \end{align}$$Here is a list of the numbers from 28247 (my diurnal age tomorrow) up to 40000:

28247, 28253, 28261, 28291, 28331, 28337, 28367, 28417, 28423, 28459, 28481, 28529, 28667, 28673, 28709, 28733, 28757, 28801, 28811, 28841, 28891, 28907, 28937, 28939, 28943, 28967, 28969, 28991, 28999, 29083, 29089, 29093, 29111, 29143, 29149, 29177, 29299, 29317, 29321, 29329, 29353, 29369, 29431, 29441, 29479, 29503, 29507, 29521, 29539, 29591, 29621, 29657, 29677, 29713, 29747, 29773, 29797, 29839, 29849, 29893, 29929, 29987, 30001, 30031, 30077, 30127, 30157, 30179, 30221, 30227, 30263, 30299, 30301, 30343, 30353, 30409, 30461, 30463, 30533, 30571, 30607, 30629, 30647, 30691, 30739, 30761, 30791, 30799, 30847, 30857, 30913, 30917, 30929, 30967, 30973, 31067, 31103, 31111, 31133, 31201, 31313, 31373, 31429, 31439, 31457, 31459, 31483, 31529, 31597, 31621, 31631, 31673, 31693, 31711, 31747, 31777, 31819, 31831, 31853, 31861, 31877, 31897, 31919, 31921, 31937, 31949, 32041, 32101, 32111, 32167, 32171, 32231, 32239, 32243, 32267, 32273, 32281, 32287, 32293, 32387, 32399, 32471, 32477, 32489, 32639, 32651, 32677, 32699, 32701, 32723, 32737, 32743, 32761, 32807, 32863, 32899, 32927, 32951, 33017, 33043, 33067, 33079, 33127, 33193, 33217, 33221, 33239, 33251, 33307, 33389, 33401, 33421, 33443, 33463, 33499, 33571, 33661, 33689, 33743, 33763, 33793, 33841, 33877, 33907, 33953, 33973, 33989, 33991, 34079, 34081, 34093, 34117, 34121, 34163, 34189, 34219, 34271, 34291, 34387, 34393, 34399, 34417, 34547, 34553, 34571, 34579, 34609, 34633, 34637, 34717, 34733, 34777, 34789, 34873, 34889, 34921, 34927, 34933, 34943, 34973, 34987, 35033, 35093, 35137, 35183, 35209, 35219, 35237, 35263, 35297, 35303, 35341, 35359, 35383, 35389, 35459, 35473, 35549, 35561, 35587, 35611, 35647, 35657, 35663, 35669, 35687, 35701, 35723, 35741, 35767, 35773, 35813, 35881, 35891, 35909, 35939, 35941, 35947, 35953, 35957, 36031, 36077, 36079, 36089, 36119, 36121, 36143, 36167, 36199, 36203, 36233, 36331, 36347, 36359, 36367, 36391, 36403, 36437, 36481, 36503, 36521, 36557, 36581, 36623, 36679, 36689, 36727, 36851, 36853, 36863, 36937, 36977, 36989, 37031, 37069, 37109, 37127, 37211, 37229, 37249, 37327, 37351, 37391, 37399, 37459, 37487, 37523, 37601, 37627, 37669, 37711, 37753, 37769, 37801, 37819, 37823, 37837, 37867, 37883, 37901, 37931, 37937, 37943, 37969, 37979, 37981, 38021, 38041, 38089, 38107, 38117, 38141, 38173, 38191, 38209, 38243, 38263, 38293, 38323, 38417, 38429, 38477, 38497, 38513, 38527, 38551, 38581, 38587, 38681, 38741, 38761, 38771, 38807, 38809, 38827, 38869, 38881, 38911, 38957, 38963, 38989, 38999, 39007, 39037, 39059, 39073, 39131, 39167, 39173, 39187, 39197, 39283, 39311, 39337, 39379, 39407, 39433, 39539, 39577, 39617, 39647, 39653, 39691, 39707, 39713, 39751, 39757, 39803, 39811, 39881, 39913, 39943, 39947

It was quite serendipitous that I thought about these sorts of numbers because, as I said, I'm 28247 days old tomorrow and this number has the property that:$$ \begin{align} 28247 &= 47 \times 601 \\47 &\rightarrow 74 =2 \times 37 \\ 601 &\rightarrow 106 =2 \times 53 \end{align}$$From my previous examples, it might look as if the two factors are having their digits reversed but remember the digits are being rearranged and not necessarily reversed. Take 28253 as an example:$$ \begin{align} 28253 &= 19 \times 1487 \\ 19 &\rightarrow 91=7 \times 13\\ 1487 &\rightarrow 7841 \text{ which is prime} \\1487 &\rightarrow 7814 = 2 \times 3907 \end{align}$$Of course this idea can be extended to sphenic numbers where each the three factors have digits that can be rearranged to form three new sphenic numbers. However, these numbers are comparatively speaking rather large. The first is 1113121 (permalink):$$ \begin{align} 1113121 &= 101 \times 103 \times 107 \\ 101&\rightarrow 110 =2 \times 5 \times 11 \\ 103 &\rightarrow 130 =2 \times 5 \times 13 \\ 107 &\rightarrow 170 = 2 \times 5 \times 17 \end{align}$$Gemini gives a good explanation of why the numbers need to be so large (link).

Sphenic Number Chains

My previous post on the topic of chains of semiprimes in arithmetic progression prompted me to investigate similar chains formed by sphenic numbers. This time we are looking for the smallest sphenic number that is at the end of an arithmetic progression of \(n\) sphenic numbers as \(n\) ranges from 1 upwards. The result for \(n\) up to 18 is as follows (permalink):

30, 42, 102, 138, 174, 442, 1010, 2278, 2422, 6494, 10322, 10586, 12694, 21434, 28466, 56426, 62902, 145930

Let's look at 28466 that is at the end of a chain of 15 sphenic numbers with a common difference of 96 (permalink):

Arithmetic Progression of 15 Sphenic Numbers

Common Difference: 96

-------------------------------------------------------

Term   | Sphenic Number   | Factorisation

-------------------------------------------------------

1      | 27122            | 2 x 71 x 191

2      | 27218            | 2 x 31 x 439

3      | 27314            | 2 x 7 x 1951

4      | 27410            | 2 x 5 x 2741

5      | 27506            | 2 x 17 x 809

6      | 27602            | 2 x 37 x 373

7      | 27698            | 2 x 11 x 1259

8      | 27794            | 2 x 13 x 1069

9      | 27890            | 2 x 5 x 2789

10     | 27986            | 2 x 7 x 1999

11     | 28082            | 2 x 19 x 739

12     | 28178            | 2 x 73 x 193

13     | 28274            | 2 x 67 x 211

14     | 28370            | 2 x 5 x 2837

15     | 28466            | 2 x 43 x 331

-------------------------------------------------------

Other tables can be generated for the other values of \(n\) but the above table is the most relevant because it covers numbers (28274, 28370 and 28466) that are upcoming for me in terms of my diurnal age.

Here are the results for 16 sphenic numbers in arithmetic progression:

Arithmetic Progression of 16 Sphenic Numbers

Common Difference: 708

-------------------------------------------------------

Term   | Sphenic Number   | Factorisation

-------------------------------------------------------

1      | 45806            | 2 x 37 x 619

2      | 46514            | 2 x 13 x 1789

3      | 47222            | 2 x 7 x 3373

4      | 47930            | 2 x 5 x 4793

5      | 48638            | 2 x 83 x 293

6      | 49346            | 2 x 11 x 2243

7      | 50054            | 2 x 29 x 863

8      | 50762            | 2 x 17 x 1493

9      | 51470            | 2 x 5 x 5147

10     | 52178            | 2 x 7 x 3727

11     | 52886            | 2 x 31 x 853

12     | 53594            | 2 x 127 x 211

13     | 54302            | 2 x 19 x 1429

14     | 55010            | 2 x 5 x 5501

15     | 55718            | 2 x 13 x 2143

16     | 56426            | 2 x 89 x 317

-------------------------------------------------------

Numbers Within Numbers

Let's formalise the concept of internal digits. Take a number like 28211. It's prime but if we remove the leftmost digits (2) and the rightmost digit (1), we are left with the number 821. This is the number within a number and it happens to be prime also. It is thus a member of OEIS A069686:

 A069686: primes whose internal digits form a prime.

The members of this sequence from 28000 to 40000 are (permalink):

28097, 28099, 28111, 28211, 28219, 28277, 28279, 28297, 28393, 28537, 28571, 28573, 28579, 28591, 28597, 28631, 28771, 28813, 28817, 28837, 28871, 28879, 29077, 29191, 29297, 29411, 29473, 29531, 29537, 29671, 29717, 29833, 29837, 29917, 30029, 30059, 30071, 30113, 30119, 30133, 30137, 30139, 30197, 30293, 30313, 30319, 30431, 30539, 30593, 30671, 30677, 30713, 30839, 30893, 30971, 30977, 31013, 31019, 31033, 31039, 31079, 31091, 31139, 31271, 31277, 31319, 31379, 31391, 31393, 31397, 31511, 31513, 31517, 31573, 31793, 31799, 31817, 31973, 31991, 32117, 32119, 32233, 32237, 32297, 32299, 32411, 32413, 32573, 32579, 32633, 32693, 32713, 32717, 32719, 32771, 32779, 32831, 32833, 32839, 32933, 32939, 33071, 33073, 33113, 33119, 33179, 33311, 33317, 33377, 33479, 33493, 33533, 33599, 33679, 33739, 33791, 33797, 33893, 34019, 34211, 34213, 34217, 34313, 34319, 34337, 34439, 34499, 34613, 34631, 34673, 34679, 34871, 34877, 34913, 34919, 35099, 35419, 35573, 35771, 35879, 35933, 35993, 35999, 36011, 36013, 36017, 36073, 36131, 36137, 36191, 36313, 36319, 36433, 36473, 36479, 36599, 36739, 36779, 36833, 36913, 36919, 37013, 37019, 37097, 37199, 37273, 37277, 37337, 37339, 37397, 37511, 37517, 37571, 37573, 37579, 37619, 37691, 37693, 37699, 37871, 37879, 38113, 38119, 38219, 38231, 38237, 38239, 38273, 38299, 38393, 38593, 38639, 38833, 38839, 38873, 39079, 39113, 39119, 39191, 39199, 39293, 39371, 39373, 39419, 39671, 39679, 39719, 39779, 39839, 39971, 39979

We can modify the algorithm to search for square numbers within square numbers. For example, \(144 = 12^2\) and its internal digit \(4 = 2^2\) and so it is a member of OEIS A069701:

 A069701: squares with internal digits also forming a square > 0.

The members up to 40000 are few and far between: 

144, 196, 441, 841, 1369, 3364, 4096, 5041, 8649, 10816, 11449, 20164, 38416.

So the takeaway from this is that every \(n\) digit number has an (\(n-2\)) digit number inside of it provided \(n>2\) and sequences can be developed by relating this internal number to the larger number of which it forms a part.

Let's consider numbers that are biprime and whose internal digits also form a biprime. Take 28189 as an example:$$ \begin{align} 28189 &= 7 \times 4027 \\ 818 &= 2 \times 409 \end{align}$$These sorts of numbers are not listed in the OEIS. 

Similarly for triprimes. Take 28055 as an example:$$ \begin{align} 28055 &= 5 \times 31 \times  181\\805 &= 5 \times 7 \times 23 \end{align}$$Of course we could consider numbers within numbers within numbers etc. but given that I'm mainly examining five digit numbers less than 40000, I'll leave off at numbers within numbers. Of course, these relationships between numbers and their respective internal numbers only apply with certainty in the number base under consideration (base 10 so far in this post). Take our earlier example of 28211 and its internal number of 821. Both are prime. Let's change to base 9:$$ \begin{align} 28211_{10} &= 42625_9 \rightarrow 265_9 \text{ as its internal number} \\ 265_9 &= 218_{10} \text{ which is clearly not prime} \end{align} $$Primeness is base independent and if a number is not prime in a certain base then it is not prime in any base. However, the inner number while not prime in base 10 may be prime if regarded as a number in another base. Let's illustrate this with 28019. It is a prime in base 10 where the inner number, 801, is divisible by 3. However, we can note the following:$$ \begin{align} 801_{12} &= 1153_{10} \text{ which is prime}\\801_{15} &= 1801_{10} \text{ which is prime} \end{align} $$I've incorporated the search for the following types of numbers into my number analysis algorithm so that the following types of numbers can be identified:

• prime number whose internal number is also prime in any base from 2 to 16
• biprime number whose internal number is also prime using base 10 only
• triprime number whose internal number is also triprime using base 10 only
• square number whose internal number is also square using base 10 only

Concatenating Reversible Sphenic Numbers

My previous post Concatenating Emirpimes suggested the concatenation of reversible sphenic numbers as a logical sequel. A sphenic number that is still sphenic when its digits are reversed is sometimes called a \( \textbf{cinephs} \). Let's take 418 as an example of the type of number that we want to identify:$$ \begin{align} 418 &= 2 \times 11 \times 19 \\ 814 &= 2 \times 11 \times 37 \\ 418814 &= 2 \times 11 \times 19037 \end{align}$$There are, coincidentally, 418 such numbers in the range up to 40000. Here are the details (permalink):

=== Cineph Concatenation Statistics ===

Range evaluated: 1 to 40000

1. Sphenic numbers: 7720 (19.30% of the range)

2. Cinephs: 1890 (4.72% of the range)

3. Successful Cinephs: 418 (1.0450% of the range)

=== Comma-Separated List of Successful Cinephs ===

418, 682, 759, 814, 957, 1034, 1095, 1113, 1185, 1342, 1419, 1446, 1606, 1614, 1902, 1965, 2014, 2035, 2282, 2414, 2431, 2438, 2494, 2635, 2665, 2686, 3018, 3059, 3138, 3201, 3278, 3297, 3358, 3495, 3597, 3606, 3678, 3685, 3714, 3729, 3765, 3926, 4301, 4382, 4543, 4565, 4715, 4945, 5174, 5362, 5495, 5863, 5986, 6035, 6061, 6083, 6206, 6293, 6806, 7185, 7305, 7414, 7449, 7567, 7645, 7657, 7662, 7718, 8155, 8174, 8378, 8386, 8390, 8393, 8555, 8734, 8786, 8789, 8987, 9138, 9141, 9213, 9309, 9321, 9354, 9362, 9503, 9515, 9519, 9807, 9822, 9885, 9951, 9978, 10095, 10246, 10263, 10274, 10326, 10382, 10490, 10502, 10509, 10554, 10635, 10761, 10835, 10879, 10915, 11098, 11342, 11395, 11398, 11407, 11577, 11605, 11753, 11922, 11937, 11958, 11982, 12174, 12218, 12257, 12378, 12498, 12529, 12551, 12595, 12859, 12874, 12890, 12914, 13137, 13143, 13222, 13305, 13514, 13574, 13618, 13634, 13783, 13786, 13906, 14035, 14043, 14055, 14127, 14174, 14223, 14298, 14313, 14443, 14573, 14590, 14619, 14646, 14655, 14846, 14889, 14894, 14955, 15235, 15323, 15422, 15430, 15521, 15585, 15626, 15657, 15794, 15818, 15914, 15958, 15994, 16098, 16159, 16518, 16558, 16666, 16701, 16914, 16915, 16946, 17006, 17139, 17353, 17445, 17518, 17534, 17799, 17805, 17814, 17818, 17978, 18093, 18123, 18147, 18205, 18222, 18231, 18309, 18395, 18458, 18555, 18579, 18685, 18717, 18814, 18854, 19023, 19178, 19221, 19317, 19382, 19545, 19605, 19623, 19877, 19923, 19947, 19987, 20009, 20066, 20195, 20270, 20305, 20315, 20530, 20594, 20657, 20726, 20758, 20870, 20905, 20945, 20965, 20966, 22058, 22282, 22562, 22582, 22591, 22645, 22933, 22939, 22945, 22970, 24002, 24026, 24095, 24265, 24409, 24479, 24583, 24590, 24739, 24817, 24878, 24962, 24973, 26090, 26102, 26134, 26146, 26245, 26266, 26273, 26429, 26494, 26506, 26546, 26555, 26614, 26638, 26663, 26758, 26873, 26878, 26930, 26939, 26990, 28085, 28165, 28186, 28222, 28237, 28358, 28418, 28426, 28441, 28483, 28514, 28645, 28747, 28835, 28870, 28918, 30054, 30099, 30126, 30277, 30418, 30502, 30651, 30795, 30943, 31026, 31027, 31035, 31191, 31254, 31386, 31658, 31719, 31978, 32039, 32043, 32181, 32295, 32351, 32417, 32442, 32613, 32846, 32857, 32898, 32907, 32938, 33143, 33162, 33186, 33378, 33429, 33473, 33555, 33583, 33585, 33754, 33818, 33835, 33906, 33918, 33922, 34005, 34077, 34078, 34131, 34133, 34158, 34203, 34222, 34257, 34474, 34562, 34563, 34661, 34705, 34826, 35006, 35094, 35123, 35202, 35274, 35319, 35382, 35618, 35619, 35871, 35949, 35985, 36003, 36039, 36165, 36174, 36226, 36355, 36474, 36586, 36606, 36627, 36935, 36958, 37317, 37329, 37367, 37411, 37418, 37433, 37527, 37614, 37634, 37743, 37754, 37970, 38049, 38085, 38193, 38294, 38399, 38451, 38467, 38533, 38555, 38643, 39081, 39169, 39305, 39306, 39351, 39358, 39399, 39458, 39562, 39785, 39842, 39918, 39966

As we learned in my previous post: 

There is a fundamental rule in number theory: Every palindrome with an even number of digits is divisible by 11.

Thus all the above numbers have 11 as a prime factor. Let's take 39358 as another example: $$ \begin{align} 39358 &= 2 \times 11 \times 1789 \\ 85393 &= 7 \times 11 \times 1109 \\ 3935885393 &= 11 \times 397 \times 901279\end{align}$$Of course, if we were to try this technique on emirps (reversible primes), we would never end up with a prime because the resultant number would always be divisible by 11. For example, 37 and 73 combine to form$$ 3773 = 11 \times 343 =11 \times 7^3$$

First Fours, Threes and Twos of a Kind

The first thing we notice about \( \textbf{27807}\) is its factorisation:$$27807 = 3 \times 13 \times 23 \times 31$$This number has four distinct prime factors and all of them contain the digit 3. The number indicates my diurnal age today. How often does this occur (that a number has four distinct prime factors and all of them contain the digit 3). Well, 27807 is the first such number. Here is the list of numbers with this property up to 100,000: 27807, 33189, 38571, 44733, 47541, 51987, 62049, 64077, 65481, 74451, 76479, 79143, 88257, 88881, 91977, 92391. Table 1 shows these numbers together with their factorisations.

Table 1

A natural question to ask is what about other digits? In the range up to 100,000, there are only four numbers with four distinct prime factors all of which contain the digit 1. These are 46189, 75361, 84227 and 99671. Table 2 shows these numbers together with their factorisations.

Table 2

Apart from the digits 1 and 3, there are no other numbers in the range up to 100,000 with four distinct prime factors each of which contain the same digit. Such numbers exist of course but they are larger than 100,000. Table 3 shows the results for all the digits from 0 to 9.

Table 3: permalink

Thus we see that 27807 is unique in that it is the smallest number with four distinct prime factors such that each factor contains the same digit at least once. I'm pleased that I spotted this as it is easy to miss. We can construct a similar table for sphenic numbers as can be seen in Table 4 where the fourth factor appearing in Table 3 is omitted.

Table 4: permalink

So we see that 897 is the smallest sphenic number whose three distinct factors contain the same digit at least once. While we're here we may as well show the results for semiprimes with two distinct prime factors as well. See Table 5.

Table 5: permalink

Thus 39 is the smallest semiprime with two distinct prime factors such that each factor contains the same digit at least once. If we didn't specify distinct prime factors then 4 = 2 x 2 would win out.

Some Special Sphenic Numbers

Today I turned \( \textbf{27782} \) days old and one of the properties of 27782 is that it is sphenic since:$$27782=2 \times 29 \times 479$$However, looking at the digital roots of the number and its factor we notice an interesting fact:$$ \underbrace{27782}_{8}=\underbrace{2}_{2} \times \underbrace{29}_{2} \times \underbrace{479}_{2}$$The respective digital roots are shown under the number and its factors and we see that the digital roots of the factors (2) multiply together to give the digital root of the number (8). The only other way this can occur is if all the digital roots are 1.

In the range up to 100,000 there are 230 sphenic numbers with the property that the digital roots of the factors multiply to give the digital root of the number. These numbers are (permalink):

638, 1034, 1826, 2222, 2726, 3014, 3806, 4202, 4814, 4994, 5786, 5858, 6182, 6974, 7766, 7802, 7946, 8558, 9494, 9746, 10034, 10142, 10538, 11078, 12518, 12878, 12914, 13166, 14102, 14498, 14894, 14993, 15254, 16262, 16298, 16766, 17954, 18062, 18386, 18458, 18854, 20042, 20438, 20474, 20834, 21338, 21626, 22418, 22562, 22742, 24002, 24398, 24722, 25586, 25694, 25982, 26414, 26477, 26738, 26774, 27674, 27782, 28358, 28718, 28754, 29798, 29942, 31526, 31706, 31922, 32219, 32714, 33002, 33182, 33506, 34046, 34298, 34946, 35486, 36566, 36674, 37178, 37682, 37862, 38222, 38582, 39266, 39842, 40634, 41642, 41822, 42911, 43334, 43406, 43658, 43703, 44594, 45026, 45386, 45782, 45854, 46178, 46646, 47366, 47402, 47618, 48554, 48662, 49346, 49706, 50534, 51319, 51326, 51722, 52217, 52334, 52622, 52838, 53126, 53306, 53486, 53702, 53882, 54098, 54494, 54926, 55178, 55187, 55682, 56078, 56762, 57014, 57662, 58454, 58598, 59102, 59246, 59642, 60038, 60254, 60929, 61622, 61946, 62018, 62198, 62414, 63278, 63638, 63998, 64034, 64322, 64394, 64574, 65186, 65978, 66086, 67454, 67958, 68498, 70586, 70829, 71306, 71522, 72062, 72413, 73106, 73538, 73898, 74762, 75086, 75806, 75878, 76274, 76526, 76627, 76994, 77174, 77858, 78254, 78542, 78578, 78866, 78938, 79046, 79514, 80558, 81422, 81818, 83114, 83897, 84158, 84986, 85382, 85634, 86174, 86246, 86714, 86858, 87326, 87758, 88154, 88334, 89018, 89281, 89441, 89486, 89639, 89738, 90422, 90926, 90998, 92213, 92402, 93122, 93302, 93554, 93698, 94454, 95678, 95786, 96686, 96722, 96758, 97226, 97262, 97442, 98054, 98747, 98846, 99818

However, of these only three have digital roots that are all equal to 1. These are:$$ \begin{align} 51319 &= 19 \times 37 \times 73\\76627 &= 19 \times 37 \times 109\\ 89281 &= 19 \times 37 \times 127 \end{align} $$As can be seen, two of the factors (19 and 37) are the same for all three numbers. If we extend the range to one million, these two factor and 73 (the reversal of 37) make frequent appearances.

If we relax the requirement that the digital roots of the factors must be equal and require only the the digital roots of the factors multiply together to give the digital root of the number, then we find that 1969 numbers satisfy and that will include the 230 numbers mentioned earlier (permalink). An example would be 27813:$$ \underbrace{27813}_{3} = \underbrace{3}_{3} \times \underbrace{73}_{1} \times \underbrace{127}_{1}$$

The Prime Constant And Beyond

A recent Numberphile video informed me of the prime constant which firstly incodes all the primes within an infinitely long sequence of 0's and 1's as shown below:$$01101010001010001010001000001010000010001010001 \dots$$The leading zero corresponds to 1 which is not prime and the next two 1's correspond to 2 and 3 that are prime and so on. The next step is to add a decimal point in front of the leading zero to get:$$0.01101010001010001010001000001010000010001010001 \dots$$This represents a number between 0 and 1 and if we interpret this as a number in base 2 we have a constant that can be expressed as (permalink):$$ \begin{align} \text{Prime Constant } &= \frac{0}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4} + \dots \\ \\&\approx 0.414682509851112 \dots \end{align}$$Of course, we could equally well create the non-prime constant by representing every non-prime by a 1 and every prime as a 0. This gives:$$ \text{Non-Prime Constant } \approx 0.585317490148888 \dots$$The prime constant and the non-prime constant of course add to 1. The same idea can be applied to create other constants, for example a Fibonacci constant. The Fibonacci numbers are:$$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots $$This generates a series of 0's and 1's as follows:$$ 11101001000010000000100000000000010000 \dots$$It can be seen that the three leading 1's are there because 1, 2 and 3 are Fibonacci numbers whereas 4 is not and so it represented by a 0 etc. This gives (permalink):$$ \text{Fibonacci Constant } \approx 0.910278797207866 \dots$$It can thus be seen that all monotonically increasing sequences like the sequence of prime numbers and the sequence of Fibonacci numbers can be represented by a constant between 0 and 1. The base 2 representation is arbitrary but simple but other bases could be used especially to represent more than one sequence. For example, base 4 could be used to represent primes, square-free semiprimes and sphenic numbers. So a 3 could represent a square-free semiprime, a 2 could represent a sphenic number, a 1 could represent a prime and 0 could represent a number that is none of these. This leads to another constant, let's call it the 1-2-3 Factor Constant, where we have a series of 0's, 1's, 2's and 3's:$$011012100210122010102210020013102220 \dots$$These again are converted to a number between 0 and 1:$$0.011012100210122010102210020013102220 \dots$$Interpreting this as a number in base 4 leads to (permalink):$$ \begin{align} \text{1-2-3 Factor Constant } &= \frac{0}{4^1}+\frac{1}{4^2}+\frac{1}{4^3} + \dots \\  \\ &\approx 0.0796530489502776 \dots \end{align}$$There is endless fun to be had in generating these sorts of constants from multiple sequences.

Dancing Digits

Whenever I'm confronted with a number associated with my diurnal age that seems to have no interesting properties, I inevitably find something very special and interesting about that number. Yesterday's number, 27542, was a number of this sort and it took me a day to stumble upon what's interesting about it.

My starting point was that it's a sphenic number because:$$2542=2 \times 47 \times 293$$Such numbers can be viewed as sphenic bricks with the three prime factors corresponding to the length, width and height. The surface area of such a brick means that there is always a second number that is inextricably linked to the original sphenic number and I've written about this in earlier posts. In the case of 27542, this second number and the surface area of the brick is 28902. This second number however, is also sphenic since we have:$$28902=2 \times 3 \times 4817$$This means that we can find the surface area of this second brick. It is 48182 which is not sphenic. However, we now have a triplet of numbers formed:$$27542, 28902, 48182$$If we find the product of these three numbers, it turns out to be an interesting number:$$27542 \times 28902 \times 48182 = 38353781868888$$It's interesting because it's 14 digits long and the digit 8 comprises precisely half of them.

The question then is how common is it for such triplets of numbers, when multiplied, to generate a number in which a single digit comprises at least 50% of all the digits? Let's reflect on the criteria for such numbers:

• the number must be sphenic and constitutes the first sphenic brick: p
• the surface area of this brick must also be a sphenic number: q
• this second number constitutes the second sphenic brick
• the surface area of this second brick constitutes the third number: r
• the product of p, q and r must contain a digit that comprises at least 50% of the digits of the number.

In the case of the digit 8, there are only three other numbers that qualify in the range up to 100,000 and these can be viewed in Figure 1. The first number in the list is 27542.

Figure 1: plethora of the digit 8

So it turns out that 27542 is the first member of a rather special sequence indeed. What about other digits? Let's start with 0.  Figure 2 shows the results for the digit 0, again up to 100,000.

Figure 2: plethora of the digit 0

The results for the digit 1 are shown in Figure 3.

Figure 3: plethora of the digit 1

For digits 2 and 3 there are no numbers and the results for digit 4 are shown in Figure 4.

Figure 4: plethora of the digit 4

For digit 6, 7 and 9 only one result is found in each case. See Figures 5, 6, 7 and 8.

Figure 5: plethora of the digit 5

Figure 6: plethora of the digit 6

Figure 7: plethora of the digit 7

Figure 8: plethora of the digit 9

Here is a permalink to the algorithm used to generate these numbers. Overall then, the numbers which produce a single digit that accounts for at least 50% of the final product of digits are:

1833, 1887, 7189, 14833, 15589, 16242, 16405, 27542, 36449, 38006, 38319, 43589, 87731

A very exclusive club indeed. Of course these number properties are base-dependent and so  fall into the realm of recreational mathematics but numberphiles are indifferent to such divisions and simply delight in the dance of the digits.

Some Special Sphenic Numbers

What struck me about the number associated with my diurnal age today was that it is sphenic and all three factors as well as the number itself share one digit in common, namely the digit 1. The number is:$$27511=11 \times 41 \times 61$$This got me wondering how many sphenic numbers in the range up to 40,000 have this property. Well it turns out that 78 numbers do. These numbers are (permalink):

1

2431, 2717, 4199, 6851, 9061, 10013, 10127, 10153, 11407, 12749, 13243, 13277, 13481, 13981, 14443, 14729, 14839, 15067, 15301, 15587, 15691, 16159, 16523, 17537, 18161, 18733, 18887, 19261, 19591, 19703, 19877, 20801, 21109, 21131, 21307, 21527, 21593, 21607, 22321, 22451, 22781, 23617, 23881, 24149, 24211, 25441, 25619, 27313, 27511, 27911, 28171, 28613, 28951, 29051, 30173, 30481, 30719, 31141, 31229, 31369, 31559, 32021, 32147, 32351, 32513, 32813, 33371, 34441, 34561, 35123, 35717, 36091, 36157, 37169, 37213, 37411, 37417, 39919

Naturally, I decided to investigate the remaining digits from 2 to 9. Here is what I found. 

2

For the digit 2, there are 22 numbers that are sphenic and in which all three factors as well as the number itself share the digit 2 in common. The first such number is 5842:$$5842=2 \times 23 \times 127$$These numbers are (permalink):

5842, 10258, 10442, 11822, 12098, 12238, 12374, 12466, 12742, 12926, 12934, 13282, 13862, 15254, 15602, 16298, 23966, 24058, 24418, 30218, 33442, 38042

3

For the digit 3, there are 138 numbers that are sphenic and in which all three factors as well as the number itself share the digit 3 in common. The first such number is:$$1443=3 \times 13 \times 37$$These numbers are (permalink):

1443, 2139, 2553, 3237, 3441, 3657, 3999, 4773, 5037, 5343, 5883, 6357, 6837, 8103, 9039, 9213, 9321, 9453, 11037, 11063, 11433, 11937, 11973, 12183, 12363, 12543, 13143, 13197, 13287, 13317, 13533, 13611, 13767, 14313, 14937, 15387, 15483, 16377, 17329, 17673, 17931, 18093, 19203, 20397, 20683, 20739, 21183, 21359, 21423, 21783, 21873, 22317, 22839, 23127, 23253, 23907, 23943, 24357, 24753, 25323, 25493, 25737, 25863, 26319, 26381, 26637, 27393, 28137, 28923, 29193, 29739, 30003, 30057, 30147, 30291, 30441, 30567, 30659, 30687, 30783, 30797, 30831, 31341, 31413, 31947, 32097, 32271, 32457, 32523, 32619, 32721, 32829, 33267, 33387, 33449, 33657, 33787, 33927, 34077, 34113, 34131, 34437, 34521, 34611, 34689, 34707, 34743, 34917, 35113, 35187, 35247, 35457, 35619, 35697, 36087, 36177, 36507, 36543, 36593, 36741, 36921, 37047, 37167, 37407, 37789, 37797, 37887, 38001, 38337, 38517, 38739, 38847, 39169, 39183, 39507, 39603, 39849, 39923

4 to 9

There are no numbers in the range up to 40,000 that are sphenic and in which all three factors as well as the number itself share the digit 4 in common. If we consider the range up to one million, we find 20 numbers. The first of these is:$$424883 = 41 \times 43 \times 241$$In the range up to 40,000, there is only one number that is sphenic and in which all three factors as well as the number itself share the digit 5 in common. This is the number:$$15635 = 5 \times 53 \times 59$$There are no numbers in the range up to 40,000 that are sphenic and in which all three factors as well as the number itself share the digit 6 in common. If we consider the range up to one million, we find two numbers. The first of these is:$$666181 = 61 \times 67 \times 163$$There are 14 numbers that are sphenic and in which all three factors as well as the number itself share the digit 7 in common. The first such number is:$$7973 = 7 \times 17 \times 67$$These numbers are (permalink):

7973, 8687, 12173, 12733, 17353, 18907, 19873, 20587, 24017, 27013, 27713, 34237, 37051, 37723

For the digit 8, there are no sphenic numbers that satisfy even in the range up to one million. For the digit 9, there is only one number in the range up to 40,000 that satisfies and that is:$$32509 = 19 \times 29 \times 59$$Before leaving, I'll return to the number that started all this: 27511. It has some other interesting properties involving prime numbers. These are:

• number + sum of digits is prime: 27511 + 16 = 27527
• number + product of digits is prime: 27511 + 70 = 27581
• concatenation of prime factors in ascending order is prime: 114161
• concatenation 116141 is also prime

The algorithm used earlier can be easily modified (permalink) to accommodate a number of distinct prime factors other than 3. In the case of four distinct prime factors, it is only the digit 3 that yields any numbers in the range up to 40,000. These numbers are:$$ \begin{align} 33189 &= 3 \times 13 \times 23 \times 37 \\38571 &= 3 \times 13 \times 23 \times 43 \end{align} $$Once the range is extended to one million, the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 have 132, 11, 277, 0, 0, 0, 22, 0 and 0 corresponding numbers respectively.

A Multiplicity of Digits: Part 2

A variation on the theme of my previous post, that also involves the multiple occurrence of the same digits, are these numbers that comprise a sequence that I've referenced as S107 in my Bespoken for Sequences database. 

Sphenic numbers containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first example of such a number is 1443 = 3 * 13 * 37 with a digital root of 3. There are 61 such numbers in the range up to 40000. Here is the list (permalink):

1443, 3441, 3657, 3999, 4773, 6357, 8103, 9039, 9453, 11037, 11433, 11937, 11973, 13143, 13197, 13287, 13611, 14313, 15483, 17931, 18093, 20397, 20739, 21423, 21783, 21873, 23907, 23943, 24357, 24753, 26319, 28137, 29739, 30441, 30567, 30783, 31341, 31413, 32097, 32457, 32619, 33267, 34077, 34113, 34131, 34437, 34689, 34707, 34743, 35247, 35697, 36507, 36543, 36741, 36921, 37047, 37407, 38001, 38739, 38847, 39603

A twist on this theme is to consider sphenic numbers that do NOT contain the digit 3. Such numbers could be considered as having a hidden multiplicity of digits because the prevalence of the digit is not immediately obvious. The same could be said of the numbers just mentioned but those cases the repeating digit is overtly visible. Here is the revised criteria:

Sphenic numbers NOT containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first such number is 1209 = 3 * 13 * 31 with a digital root of 3. There are 45 such numbers in the range up to 40000. Here they are (permalink):

1209, 1677, 2847, 4017, 5421, 5727, 6789, 7527, 7797, 8697, 9417, 9579, 12207, 12909, 12927, 14547, 14781, 15159, 15429, 16077, 16491, 16887, 17121, 17949, 17967, 18057, 18147, 20217, 20829, 21027, 21459, 22557, 24609, 24771, 24897, 25077, 26247, 26427, 26841, 27507, 28551, 28587, 28767, 28821, 29109

Rather than sphenic numbers, with three distinct prime factors, we could consider biprimes or numbers with two distinct prime factors. Firstly let's look at numbers with properities as follows:

Biprimes containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 73 such numbers in the range up to 40000. The first of these is 254 = 2 * 127 with a digital root of 2. Here they are (permalink):

254, 542, 2558, 2594, 2846, 3242, 4286, 4322, 4502, 4682, 5042, 5267, 5294, 5582, 5942, 8462, 12242, 12422, 12458, 12494, 12854, 14258, 16526, 17246, 17642, 18254, 18929, 19442, 20486, 21458, 22502, 22574, 23483, 23654, 24014, 24041, 24086, 24194, 24482, 24554, 24842, 24914, 25022, 25094, 25166, 25202, 25238, 25274, 25526, 25562, 25598, 25706, 25778, 25814, 25958, 25967, 26498, 26534, 27254, 27623, 28442, 28451, 28586, 28829, 29693, 29846, 30242, 30521, 32546, 33842, 35246, 36254, 39629

Again we can consider the revised criteria:

Biprimes NOT containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 53 such numbers in the range up to 40000 with the first being 1046 = 2 * 523 with a digital root of 2. Here are the numbers (permalink):

1046, 1658, 3683, 4034, 4106, 4178, 4358, 4538, 4574, 4754, 4853, 4934, 5006, 5078, 5114, 5186, 5366, 5438, 5834, 5906, 6509, 6518, 7058, 7454, 7859, 10046, 11054, 11846, 13646, 14438, 14474, 15167, 15446, 16418, 17858, 19658, 30854, 31646, 33401, 34058, 34418, 34598, 35687, 35858, 36434, 36506, 36578, 37046, 37091, 37613, 38414, 38846, 39854