Supercharged Semiprimes

 There is a set of non-square semiprimes defined by two criteria:

• the two prime factors concatenate in either order to form two new semiprimes
• the digits of the two prime factors can each be rearranged to form semiprimes

28291 is an example of such a semiprime. Let's examine its properties where || represents concatenation:$$ \begin{align} 28291 &=19 \times 1489\\ 19 \, || \, 1489 &= 191489 \\ &= 53 \times 3613 \\ 1489 \, || \,19  &=148919 \\ &= 137 \times 1087 \\ 19 &\rightarrow 91 \\&=7 \times 13 \\1489 &\rightarrow 1894 \\ &= 2 \times 947 \end{align}$$Up to 40000, the semiprimes that meet both criteria are (permalink):

779, 817, 1121, 1763, 1957, 2071, 2869, 3173, 3403, 3629, 4579, 4687, 4897, 5149, 5977, 6973, 7009, 7181, 7261, 7367, 7493, 8077, 8341, 8413, 8549, 8851, 8977, 9071, 9937, 10123, 10363, 10393, 10679, 10697, 10951, 11051, 11419, 11647, 12407, 12599, 13207, 14317, 14473, 14719, 14809, 14953, 15007, 15143, 15529, 15637, 15751, 16283, 16297, 16321, 16351, 16789, 16873, 17671, 18391, 18563, 18643, 19177, 19247, 19451, 20081, 20311, 20653, 20729, 20989, 21071, 21223, 21733, 21829, 21887, 21971, 22313, 22361, 22489, 22819, 22837, 22879, 22987, 23083, 23351, 23521, 24257, 24377, 24559, 24751, 24757, 24823, 25061, 25843, 25901, 26179, 26239, 26617, 26671, 26969, 27089, 27161, 27199, 27331, 27383, 27589, 27913, 28177, 28291, 28801, 28907, 28937, 28999, 29149, 29329, 29621, 30301, 30571, 30847, 31111, 31921, 32101, 32239, 32293, 32387, 32651, 32699, 33307, 33907, 33991, 34093, 34571, 34579, 34633, 34873, 34927, 35137, 35209, 35341, 35389, 35587, 35701, 35881, 36031, 36199, 36689, 37069, 37127, 37867, 37901, 38021, 38141, 38173, 38323, 38477, 38497, 38989, 39187, 39433, 39707, 39757

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.

ODD - and EVEN + Improved Format

Some time ago I got Gemini to create a 197 page document that identifies attractors and vortices arising from the recursive ODD - and EVEN + algorithm. It also lists their number of captives. Today I got Gemini to summarise and reformat this information so that it is more readable using the exact same template it created for my August 15th post titled ODD + and EVEN - Improved Format. That 142 page document can be located here. Here are some excerpts:

Figure 1

Figure 2

Table 1

Table 2

Table 3

Table 4

Notice that while 8987 has the record number of captives under the ODD - and EVEN + recursive algorithm (in the range up to 40000), it has ZERO captives under the ODD + and EVEN - recursive algorithm. Conversely, while 38013 is a vortical in the mighty vortex {38013, 38012, 38006, 37995, 38028} with 564 captives under the ODD + and EVEN - recursive algorithm, it is a mere captive of the attractor 38050 under the ODD - and EVEN + recursive algorithm.

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).

Pronic Determinants of Circulant Matrices

Consider the number 28235 that is my diurnal age today. It has a circulant matrix with a determinant 10100 that is a pronic number since 10100 = 100 x 101.$$\begin{bmatrix}

2 & 8 & 2 & 3 & 5 \\

8 & 2 & 3 & 5 & 2 \\

2 & 3 & 5 & 2 & 8 \\

3 & 5 & 2 & 8 & 2 \\

5 & 2 & 8 & 2 & 3

\end{bmatrix}$$What's interesting is that most of the permutations of the digits of 28235 have determinants of their circulant matrices that are also pronic (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22358        | 15500           | 124 x 125      
22385        | 10100           | 100 x 101      
22538        | 19100           |                
22583        | 10100           | 100 x 101      
22835        | 19100           |                
22853        | 15500           | 124 x 125      
23258        | 19100           |                
23285        | 19100           |                
23528        | 10100           | 100 x 101      
23582        | 15500           | 124 x 125      
23825        | 15500           | 124 x 125      
23852        | 10100           | 100 x 101      
25238        | 15500           | 124 x 125      
25283        | 15500           | 124 x 125      
25328        | 10100           | 100 x 101      
25382        | 19100           |                
25823        | 19100           |                
25832        | 10100           | 100 x 101      
28235        | 10100           | 100 x 101      
28253        | 10100           | 100 x 101      
28325        | 15500           | 124 x 125      
28352        | 19100           |                
28523        | 19100           |                
28532        | 15500           | 124 x 125      
32258        | 10100           | 100 x 101      
32285        | 15500           | 124 x 125      
32528        | 15500           | 124 x 125      
32582        | 19100           |                
32825        | 10100           | 100 x 101      
32852        | 19100           |                
35228        | 19100           |                
35282        | 10100           | 100 x 101      
35822        | 15500           | 124 x 125      
38225        | 19100           |                
38252        | 15500           | 124 x 125      
38522        | 10100           | 100 x 101      
52238        | 10100           | 100 x 101      
52283        | 19100           |                
52328        | 19100           |                
52382        | 15500           | 124 x 125      
52823        | 10100           | 100 x 101      
52832        | 15500           | 124 x 125      
53228        | 15500           | 124 x 125      
53282        | 10100           | 100 x 101      
53822        | 19100           |                
58223        | 15500           | 124 x 125      
58232        | 19100           |                
58322        | 10100           | 100 x 101      
82235        | 15500           | 124 x 125      
82253        | 19100           |                
82325        | 19100           |                
82352        | 10100           | 100 x 101      
82523        | 15500           | 124 x 125      
82532        | 10100           | 100 x 101      
83225        | 10100           | 100 x 101      
83252        | 15500           | 124 x 125      
83522        | 19100           |                
85223        | 10100           | 100 x 101      
85232        | 19100           |                
85322        | 15500           | 124 x 125  

Note that it is only when the determinant is 19100 that it is not pronic since 19100 = 100 x 191. In my post titled Determinants of Circulant Matrices, I listed all numbers up to 40000 with the property that the determinants of their circulant matrices were pronic. The numbers between 28000 and 40000 are:

28235, 28253, 28325, 28327, 28453, 28479, 28532, 28543, 28574, 28619, 28732, 28776, 29054, 29168, 29245, 29254, 29555, 29700, 29748, 30171, 30179, 30566, 30575, 30665, 30900, 31100, 31107, 31134, 31233, 31323, 31332, 31355, 31358, 31385, 31400, 31413, 31422, 31440, 31510, 31637, 31646, 31684, 31763, 31907, 32124, 32133, 32223, 32232, 32241, 32258, 32285, 32287, 32313, 32322, 32331, 32528, 32728, 32825, 32845, 32854, 32960, 33123, 33132, 33141, 33176, 33213, 33222, 33231, 33312, 33321, 33335, 33353, 33515, 33518, 33533, 33569, 33671, 33789, 33815, 33965, 33987, 34100, 34166, 34212, 34258, 34311, 34410, 34582, 34599, 34700, 34861, 35153, 35183, 35248, 35282, 35333, 35482, 35507, 35531, 35606, 35693, 35822, 35831, 35936, 35949, 35996, 35999, 36056, 36065, 36137, 36359, 36395, 36418, 36461, 36506, 36614, 36713, 36920, 36995, 37011, 37055, 37091, 37316, 37361, 37400, 37700, 37799, 37822, 37893, 37938, 37979, 38146, 38153, 38252, 38272, 38379, 38397, 38425, 38522, 38524, 38531, 39495, 39536, 39569, 39599, 39653, 39659, 39738, 39797, 39873, 39900, 39954, 39959, 39977, 39995

Note that with 28235 and its digit permutations there are TWO determinants that satisfy. These are:

• \(10100 = 100 \times 101\)
• \(15500 = 124 \times 125\)

This is generally not the case. Consider 28327 and its digit permutations where only the determinant 14762 = 121 x 122 satisfies (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22378        | 14762           | 121 x 122      
22387        | 11462           |                
22738        | 27962           |                
22783        | 11462           |                
22837        | 27962           |                
22873        | 14762           | 121 x 122      
23278        | 27962           |                
23287        | 27962           |                
23728        | 11462           |                
23782        | 14762           | 121 x 122      
23827        | 14762           | 121 x 122      
23872        | 11462           |                
27238        | 14762           | 121 x 122      
27283        | 14762           | 121 x 122      
27328        | 11462           |                
27382        | 27962           |                
27823        | 27962           |                
27832        | 11462           |                
28237        | 11462           |                
28273        | 11462           |                
28327        | 14762           | 121 x 122      
28372        | 27962           |                
28723        | 27962           |                
28732        | 14762           | 121 x 122      
32278        | 11462           |                
32287        | 14762           | 121 x 122      
32728        | 14762           | 121 x 122      
32782        | 27962           |                
32827        | 11462           |                
32872        | 27962           |                
37228        | 27962           |                
37282        | 11462           |                
37822        | 14762           | 121 x 122      
38227        | 27962           |                
38272        | 14762           | 121 x 122      
38722        | 11462           |                
72238        | 11462           |                
72283        | 27962           |                
72328        | 27962           |                
72382        | 14762           | 121 x 122      
72823        | 11462           |                
72832        | 14762           | 121 x 122      
73228        | 14762           | 121 x 122      
73282        | 11462           |                
73822        | 27962           |                
78223        | 14762           | 121 x 122      
78232        | 27962           |                
78322        | 11462           |                
82237        | 14762           | 121 x 122      
82273        | 27962           |                
82327        | 27962           |                
82372        | 11462           |                
82723        | 14762           | 121 x 122      
82732        | 11462           |                
83227        | 11462           |                
83272        | 14762           | 121 x 122      
83722        | 27962           |                
87223        | 11462           |                
87232        | 27962           |                
87322        | 14762           | 121 x 122   

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

Some Categories of Primes

There is a category of prime numbers with the property that when both the sum of their digits and the product of their digits is added to the number then the new, resultant numbers are also prime. An example would be 28181 with a sum of digits of 20 and a product of digits of 128 where:$$ \begin{align} 28181 + 20 &= 28201 \text{ prime} \\ 28181 + 128 &= 28309 \text{ prime} \end{align}$$In the range up to 40000, these primes have a density of 7.376% compared to all primes. Here is a list of such primes between 28000 and 40000 (permalink):

28097, 28181, 28703, 28901, 29153, 29179, 29209, 30089, 30119, 30203, 30313, 30449, 30469, 30539, 30557, 30649, 30661, 30713, 30803, 30809, 30829, 31019, 31307, 32063, 32069, 32083, 32173, 32203, 32401, 32687, 32957, 32971, 33013, 33037, 33091, 33301, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35053, 35059, 35251, 35257, 35323, 35507, 35509, 35521, 35569, 35831, 36209, 36229, 36469, 36559, 36607, 36919, 37019, 37039, 37097, 37321, 37369, 37501, 37507, 37547, 37871, 38047, 38351, 38959, 39019, 39079, 39103, 39161, 39301, 39521

These primes constitute OEIS A128717:

A128717: primes that yield another prime if one adds either the sum of its digits or the product of its digits.

Another category of prime involves its cube being pandigital, meaning that each digit from 0 to 9 occurs at least once with duplicates being permitted. Again 28181 satisfies this condition:$$28181^3 = 20753798525641$$Primes of this sort constitute:

A124629: primes \(p\) such that their cubes are pandigital.

The members of this sequence up to 40000 have a density is 1.523 % compared to all primes and these are (permalink):

5437, 6221, 7219, 8443, 10903, 11353, 15937, 17123, 18229, 19429, 20353, 20903, 20929, 21803, 21841, 21961, 22123, 22283, 22993, 23053, 23369, 23663, 24733, 25183, 25219, 25463, 26317, 26387, 26449, 27127, 27481, 28181, 28631, 28711, 28961, 29059, 29443, 29501, 30169, 31153, 31183, 32213, 32801, 33739, 33797, 33811, 33941, 34283, 35027, 35051, 35729, 35963, 36137, 36251, 36383, 36809, 36943, 37223, 37369, 37511, 37619, 37967, 38281, 38917

Another category of prime involves the average of the prime and the next prime being palindromic. Again 28181 satisfies since:$$ \frac{28181+28183}{2}=28182$$Many such primes are the lesser of a twin prime pair but not all. Primes of this sort constitute OEIS A242387:

A242387: lesser of consecutive primes whose average is a palindromic number.

The members of this sequence up to 40000 have a density of 1.213% compared to all primes and these are (permalink):

3, 5, 7, 97, 109, 281, 359, 389, 409, 509, 631, 653, 691, 743, 827, 857, 907, 937, 967, 1549, 2111, 2767, 4219, 4441, 7001, 9007, 9337, 9661, 10099, 11503, 12919, 13421, 16759, 17569, 21011, 21611, 23831, 26261, 26861, 28181, 29287, 29483, 30497, 31307, 32213, 33029, 33629, 34739, 36353, 37463, 39089

Another category of prime involves the differences between consecutive digits. Some primes have consecutive digits that differ by 6 or 7. An example is 28181 where we see that:$$ 2_{ \, 6} \, 8_{ \, 7} \, 1_{ \, 7} \, 8_{ \, 7} \, 1$$Such primes are few and far between and in the range up 40000, there are only the following:

17, 29, 71, 181, 281, 293, 607, 829, 929, 2939, 3929, 8171, 8293, 9281, 9293, 18181, 28181, 39293

Such primes belong to OEIS A048418:

A048418: primes whose consecutive digits differ by 6 or 7.

Yes another category involves totals of composite numbers between successive primes that are palindromes. 28181 qualifies once again because the next prime is its twin 28183 and the interprime number, 28182, is palindromic. Let's consider another prime, 29587. The next prime is 29599 and the composite numbers between them total 325523, a palindrome. Therefore we include 29587. These primes form OEIS A054266 with a density of only 0.8089% of the primes in the range up to 40000:

A054266: sum of composite numbers between prime \(p\) and nextprime(\(p\)) is palindromic.

The members up to 40000 are (permalink):

2, 3, 5, 109, 193, 281, 509, 661, 827, 857, 1439, 2111, 3433, 3889, 3967, 4549, 6661, 7001, 8467, 10099, 17203, 18583, 21011, 21611, 23831, 24847, 25117, 26261, 26497, 26861, 28181, 29587, 30497, 31307

We see that 28181, my diurnal age today, features in all these different categories of primes. Another category of primes (to which 28181 cannot belong) is to consider primes that only consist of non-prime digits (0, 1, 4, 6, 8 and 9). They do not contain any prime digits (2, 3, 5 or 7). Such primes belong to OEIS A034844 and comprise 5.782% of the primes up to 40000:

A034844: primes with only nonprime decimal digits.

Here are the primes up to 40000 (permalink):

11, 19, 41, 61, 89, 101, 109, 149, 181, 191, 199, 401, 409, 419, 449, 461, 491, 499, 601, 619, 641, 661, 691, 809, 811, 881, 911, 919, 941, 991, 1009, 1019, 1049, 1061, 1069, 1091, 1109, 1181, 1409, 1481, 1489, 1499, 1601, 1609, 1619, 1669, 1699, 1801, 1811, 1861, 1889, 1901, 1949, 1999, 4001, 4019, 4049, 4091, 4099, 4111, 4409, 4441, 4481, 4649, 4691, 4801, 4861, 4889, 4909, 4919, 4969, 4999, 6011, 6089, 6091, 6101, 6199, 6449, 6469, 6481, 6491, 6619, 6661, 6689, 6691, 6841, 6869, 6899, 6911, 6949, 6961, 6991, 8009, 8011, 8069, 8081, 8089, 8101, 8111, 8161, 8191, 8419, 8461, 8609, 8641, 8669, 8681, 8689, 8699, 8819, 8849, 8861, 8941, 8969, 8999, 9001, 9011, 9041, 9049, 9091, 9109, 9161, 9181, 9199, 9419, 9461, 9491, 9601, 9619, 9649, 9661, 9689, 9811, 9901, 9941, 9949, 10009, 10061, 10069, 10091, 10099, 10111, 10141, 10169, 10181, 10499, 10601, 10691, 10861, 10889, 10891, 10909, 10949, 11069, 11119, 11149, 11161, 11411, 11489, 11491, 11681, 11689, 11699, 11801, 11909, 11941, 11969, 11981, 14009, 14011, 14081, 14149, 14401, 14411, 14419, 14449, 14461, 14489, 14669, 14699, 14869, 14891, 14969, 16001, 16061, 16069, 16091, 16111, 16141, 16189, 16411, 16481, 16619, 16649, 16661, 16691, 16699, 16811, 16889, 16901, 16981, 18041, 18049, 18061, 18089, 18119, 18149, 18169, 18181, 18191, 18199, 18401, 18461, 18481, 18661, 18691, 18869, 18899, 18911, 18919, 19001, 19009, 19069, 19081, 19141, 19181, 19441, 19469, 19489, 19609, 19661, 19681, 19699, 19801, 19819, 19841, 19861, 19889, 19891, 19919, 19949, 19961, 19991

Primes beginning with 2 or 3 cannot qualify and so it is only when we reach primes beginning with 4 that membership is possible. The first of these is 40009.

We can flip this and consider only those primes that are comprised of prime digits. These form OEIS A019546:

A019546: primes whose digits are primes; primes having only {2, 3, 5, 7} as digits.

These primes have a density of 2.890% of the primes up to 40000 are they are (permalink):

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557, 5573, 5737, 7237, 7253, 7333, 7523, 7537, 7573, 7577, 7723, 7727, 7753, 7757, 22273, 22277, 22573, 22727, 22777, 23227, 23327, 23333, 23357, 23537, 23557, 23753, 23773, 25237, 25253, 25357, 25373, 25523, 25537, 25577, 25733, 27253, 27277, 27337, 27527, 27733, 27737, 27773, 32233, 32237, 32257, 32323, 32327, 32353, 32377, 32533, 32537, 32573, 33223, 33353, 33377, 33533, 33577, 33757, 33773, 35227, 35257, 35323, 35327, 35353, 35527, 35533, 35537, 35573, 35753, 37223, 37253, 37273, 37277, 37337, 37357, 37537, 37573

Highly Primeable Numbers

A composite numbers is primeable if it can be made prime by changing a single digit. If it cannot, then it is said to be unprimeable. What struck me about the number associated with my diurnal age today (28169) is how many ways (10) in which it can be made prime.

28169 is NOT unprimeable. It can be made prime with the following changes (permalink):

- Changing the '2' at position 1 (from the left) to '1' yields 18169
- Changing the '2' at position 1 (from the left) to '5' yields 58169
- Changing the '2' at position 1 (from the left) to '8' yields 88169
- Changing the '8' at position 2 (from the left) to '1' yields 21169
- Changing the '8' at position 2 (from the left) to '4' yields 24169
- Changing the '8' at position 2 (from the left) to '5' yields 25169
- Changing the '1' at position 3 (from the left) to '0' yields 28069
- Changing the '1' at position 3 (from the left) to '6' yields 28669
- Changing the '6' at position 4 (from the left) to '0' yields 28109
- Changing the '9' at position 5 (from the left) to '3' yields 28163

This got me thinking about highly primeable numbers and what numbers are record breakers by setting records for the number of ways in which they can be made prime. I got Gemini to write a program to investigate this and here is what it came up with in the range up to 100000 (permalink):

Record-Breaking Primeable Numbers

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Number          | Ways to Make Prime

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4               | 4

21              | 7

33              | 8

111             | 10

133             | 11

177             | 13

357             | 14

1001            | 15

4221            | 16

10759           | 17

11487           | 18

42189           | 20

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Here is the list of numbers: 4, 21, 33, 111, 133, 177, 357, 1001, 4221, 10759, 11487, 42189.

So we see that 28169, though highly primeable, is NOT a record breaker. 

Let's look at the 20 ways in which 42189 can be made prime:

42189 is NOT unprimeable. It can be made prime with the following changes (permalink):

 - Changing the '4' at position 1 (from the left) to '2' yields 22189
 - Changing the '4' at position 1 (from the left) to '3' yields 32189
 - Changing the '4' at position 1 (from the left) to '5' yields 52189
 - Changing the '4' at position 1 (from the left) to '6' yields 62189
 - Changing the '4' at position 1 (from the left) to '8' yields 82189
 - Changing the '4' at position 1 (from the left) to '9' yields 92189
 - Changing the '2' at position 2 (from the left) to '0' yields 40189
 - Changing the '2' at position 2 (from the left) to '1' yields 41189
 - Changing the '2' at position 2 (from the left) to '3' yields 43189
 - Changing the '2' at position 2 (from the left) to '4' yields 44189
 - Changing the '2' at position 2 (from the left) to '7' yields 47189
 - Changing the '1' at position 3 (from the left) to '0' yields 42089
 - Changing the '1' at position 3 (from the left) to '5' yields 42589
 - Changing the '1' at position 3 (from the left) to '6' yields 42689
 - Changing the '1' at position 3 (from the left) to '9' yields 42989
 - Changing the '8' at position 4 (from the left) to '3' yields 42139
 - Changing the '8' at position 4 (from the left) to '6' yields 42169
 - Changing the '8' at position 4 (from the left) to '7' yields 42179
 - Changing the '9' at position 5 (from the left) to '1' yields 42181
 - Changing the '9' at position 5 (from the left) to '7' yields 42187

While the above table shows record breakers, there are other numbers that equal existing records but do NOT set those records themselves. The following table shows these numbers (permalink) in the range up to 100,000.

Numbers Equaling an Existing Record

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Number          | Ways to Make Prime

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6               | 4

8               | 4

9               | 4

10              | 4

12              | 4

14              | 4

15              | 4

16              | 4

18              | 4

27              | 7

49              | 8

63              | 8

77              | 8

119             | 10

147             | 11

153             | 11

1011            | 15

1099            | 15

1209            | 15

1623            | 15

10637           | 16

13699           | 18

14421           | 18

16457           | 18

21717           | 18

31647           | 18

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 Here is a comma-separated list of these numbers:

6, 8, 9, 10, 12, 14, 15, 16, 18, 27, 49, 63, 77, 119, 147, 153, 1011, 1099, 1209, 1623, 10637, 13699, 14421, 16457, 21717, 31647