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

Fermat Numbers

A Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form: $$ \text{F}_n=2^{2^n}+1 $$ where n is a non-negative integer. The first few Fermat numbers are:

• \(\text{F}_0=3\)
• \(\text{F}_1=5\)
• \(\text{F}_2=17 \)
• \(\text{F}_3=257\)
• \(\text{F}_4=65537\)
• \(\text{F}_5=4294967297\)
• \(\text{F}_6=18446744073709551617\)
• \(\text{F}_7=340282366920938463463374607431768211457\)

The numbers form the initial members of sequence A000215 in the OEIS. The only known Fermat primes are the first five members of this sequence (3, 5, 17, 257 and 65537).

The number associated with my diurnal age today, \( \textbf{28289}\), is a prime and oddly it has a connection to these Fermat numbers via its membership in OEIS A229856:

   A229856: primes of the form \(384 \times k + 257\) where \(k \geq 0\)

The initial members are:

257, 641, 1409, 3329, 4481, 7937, 9473, 9857, 11393, 11777, 12161, 13313, 13697, 14081, 15233, 16001, 17921, 19073, 19457, 19841, 21377, 23297, 25601, 28289, 30593, 30977, 35201, 35969, 36353, 37889, 38273, 39041

It's stated in the comments to this OEIS sequence that:

Every Fermat number greater than 257 has a prime factor of the form \(384 \times k + 257, k > 0\).

Let's test this out with the initial Fermat numbers greater than 257. Now \( \text{F}_4=65537\) is a prime and so it is its own prime factor and we find that:$$65537 = 384 \times 170 + 257$$The next Fermat numbers listed above are all composite so we need to check their prime factors:$$ \begin{align} \text{F}_5=4294967297 &= 641 \times 6700417 \\ 641 &= 384 \times 1 + 257 \end{align}$$ $$ \begin{align} \text{F}_6=18446744073709,551,617 &=274177 × 67280421310721\\ 67280421310721 &= 384 \times 175209430496+257 \end{align}$$I won't attempt \( \text{F}_7 \) as it's too lengthy but the point is made. There's an explanation that Gemini provided as to why at least one prime factor of a Fermat number must have this form. You can check that out here.

Variations on Magnanimous Numbers

Today I'm 28285 days old and one of the properties of this number is that it's magnanimous, meaning that inserting a "+" between any two digits produces a prime. In this case we have:

• 2+8285 = 8287 is prime
  
  
• 28+285 = 313 is prime
  
  
• 282+85 = 367 is prime
  
  
• 2828+5 = 2833 is prime

I review these sorts of numbers in my post titled Magnanimous Numbers from December of 2020. This got me thinking about subtraction instead of addition or to put it formally and applying the absolute value operator:

| Minuend - Subtrahend | = | Difference |

Take as an example, the number 28041 where we have:

• | 2 - 8041 | = 8039 is prime
  
  
• | 28 - 41 | = 13 is prime
  
  
• | 280 - 41 | = 239 is prime
  
  
• | 2804 - 1 | = 2803 is prime

Here is the full list of 230 such numbers up to 40000 (permalink):

13, 14, 16, 18, 20, 24, 25, 27, 29, 30, 31, 35, 36, 38, 41, 42, 46, 47, 49, 50, 52, 53, 57, 58, 61, 63, 64, 68, 69, 70, 72, 74, 75, 79, 81, 83, 85, 86, 92, 94, 96, 97, 103, 108, 114, 118, 130, 132, 138, 154, 174, 190, 198, 207, 209, 225, 245, 263, 269, 285, 301, 310, 332, 334, 356, 370, 376, 392, 409, 421, 441, 447, 463, 465, 487, 503, 507, 518, 536, 552, 558, 601, 623, 629, 643, 665, 667, 689, 709, 710, 714, 730, 736, 754, 774, 790, 796, 801, 845, 867, 907, 912, 956, 970, 992, 1003, 1114, 1152, 1158, 1332, 1354, 1510, 1614, 1758, 1930, 1992, 1998, 2007, 2009, 2025, 2241, 2421, 2465, 2603, 2609, 2663, 2685, 2729, 2825, 2841, 2865, 3154, 3310, 3370, 3376, 3392, 3436, 3512, 3596, 3634, 3730, 3790, 3956, 3970, 4021, 4047, 4087, 4201, 4245, 4341, 4401, 4467, 4603, 4663, 4665, 4687, 4801, 4867, 5052, 5156, 5514, 5592, 5658, 5952, 5996, 6001, 6023, 6029, 6043, 6065, 6067, 6089, 6163, 6203, 6245, 6269, 6365, 6563, 6607, 6623, 6689, 6863, 7009, 7134, 7270, 7314, 7390, 7396, 7570, 7734, 7936, 8069, 8241, 8265, 8285, 8607, 8625, 8667, 8669, 8847, 8865, 8885, 9092, 9290, 9356, 9736, 9896, 9976, 9992, 10003, 11532, 11598, 11952, 15154, 17574, 20007, 20063, 20841, 22041, 22401, 24029, 24203, 24263, 24623, 26009, 26265, 27029, 28041, 33190, 37396

With multiplication we would need to add 1 or subtract 1. Let's take 28034 as an example of the former where we have:

• 2 x 8034 + 1 = 16069 is prime
  
  
• 28 x 34 + 1 = 953 is prime
  
  
• 280 x 34 + 1 = 9521 is prime
  
  
• 2803 x 4 + 1 = 11213 is prime

The 421 numbers that satisfy in the range up to 40000 are (permalink):

11, 12, 14, 16, 21, 22, 23, 25, 26, 28, 29, 32, 34, 36, 41, 43, 44, 47, 49, 52, 56, 58, 61, 62, 63, 65, 66, 67, 74, 76, 82, 85, 89, 92, 94, 98, 101, 104, 106, 112, 116, 118, 128, 136, 142, 152, 166, 178, 182, 202, 203, 205, 209, 221, 223, 229, 236, 244, 248, 254, 256, 263, 265, 274, 281, 298, 302, 306, 326, 332, 336, 352, 374, 376, 392, 394, 401, 407, 425, 428, 434, 448, 449, 484, 487, 502, 508, 512, 526, 542, 548, 556, 562, 566, 584, 586, 601, 603, 607, 616, 625, 626, 632, 647, 652, 658, 661, 663, 666, 704, 706, 718, 728, 734, 748, 766, 778, 794, 805, 812, 829, 832, 844, 865, 874, 884, 902, 934, 958, 968, 982, 992, 1001, 1004, 1006, 1018, 1052, 1106, 1108, 1156, 1162, 1178, 1192, 1226, 1228, 1256, 1312, 1352, 1448, 1498, 1568, 1616, 1658, 1708, 1726, 1756, 1862, 1886, 1982, 2002, 2003, 2009, 2023, 2026, 2065, 2074, 2086, 2176, 2204, 2216, 2221, 2228, 2243, 2281, 2306, 2336, 2384, 2386, 2443, 2524, 2576, 2645, 2648, 2686, 2716, 2774, 2776, 2849, 2876, 2998, 3002, 3006, 3034, 3074, 3136, 3152, 3224, 3244, 3302, 3404, 3412, 3526, 3542, 3566, 3574, 3592, 3704, 3734, 3932, 4001, 4007, 4034, 4043, 4078, 4102, 4142, 4168, 4234, 4265, 4274, 4304, 4402, 4423, 4447, 4504, 4685, 4724, 4807, 4825, 4918, 4982, 5008, 5036, 5066, 5162, 5198, 5206, 5302, 5344, 5366, 5506, 5518, 5534, 5542, 5558, 5594, 5608, 5612, 5708, 5726, 5786, 5918, 6001, 6003, 6007, 6056, 6076, 6112, 6166, 6178, 6182, 6245, 6263, 6268, 6382, 6412, 6443, 6452, 6467, 6518, 6536, 6601, 6605, 6607, 6632, 6667, 6766, 6802, 6902, 7004, 7006, 7094, 7118, 7126, 7244, 7384, 7424, 7516, 7606, 7706, 7886, 7918, 7948, 8005, 8042, 8056, 8186, 8201, 8249, 8261, 8266, 8324, 8332, 8462, 8474, 8516, 8536, 8542, 8602, 8662, 8794, 8912, 8942, 8984, 9002, 9034, 9112, 9128, 9158, 9254, 9278, 9428, 9494, 9502, 9532, 9652, 9704, 9748, 9932, 9992, 10004, 10018, 10028, 10136, 10312, 10336, 10448, 11102, 11276, 11482, 11552, 11578, 11608, 11662, 12026, 12206, 12656, 12728, 13256, 13312, 14002, 14098, 14158, 14548, 14758, 14798, 14812, 14968, 15208, 15856, 15952, 16142, 17126, 18116, 18536, 18736, 18962, 19012, 19402, 19768, 19832, 20002, 20021, 20254, 20849, 20866, 21778, 22178, 22616, 22778, 22948, 23834, 23876, 24043, 24118, 24334, 24424, 24824, 25228, 25298, 25366, 25556, 25844, 26005, 26516, 26665, 27028, 27356, 27784, 28034, 28516, 28874, 29708, 30274, 30362, 30662, 30704, 32024, 32066, 32234, 33106, 33182, 33784, 34492, 35206, 36676, 38032, 38362, 38512, 38734, 39532

If instead we subtract 1, we get the following 400 numbers in the range up to 40000 (permalink):

13, 14, 16, 18, 22, 23, 24, 26, 27, 29, 31, 32, 34, 36, 38, 41, 42, 43, 45, 46, 48, 54, 56, 61, 62, 63, 64, 65, 67, 68, 69, 72, 76, 81, 83, 84, 86, 89, 92, 96, 98, 103, 106, 108, 114, 124, 138, 154, 162, 168, 174, 184, 198, 203, 204, 207, 209, 212, 222, 236, 242, 264, 269, 296, 301, 302, 306, 308, 324, 334, 336, 338, 384, 386, 398, 402, 405, 406, 421, 426, 427, 441, 456, 463, 468, 492, 496, 504, 512, 522, 536, 548, 572, 588, 596, 601, 603, 604, 607, 608, 618, 629, 638, 642, 643, 664, 667, 684, 702, 706, 714, 726, 762, 784, 786, 792, 801, 803, 806, 809, 825, 834, 845, 846, 863, 876, 902, 908, 912, 948, 956, 962, 972, 986, 992, 996, 1006, 1038, 1062, 1084, 1104, 1152, 1308, 1314, 1368, 1422, 1504, 1524, 1548, 1608, 1614, 1654, 1662, 1692, 1734, 1752, 1824, 1854, 1864, 1884, 1968, 1992, 2003, 2007, 2012, 2036, 2052, 2064, 2112, 2274, 2427, 2465, 2486, 2609, 2612, 2664, 2724, 2736, 2805, 3002, 3008, 3086, 3198, 3234, 3304, 3318, 3368, 3408, 3498, 3504, 3596, 3624, 3634, 3638, 3668, 3786, 3926, 3976, 4005, 4006, 4021, 4026, 4083, 4152, 4158, 4188, 4207, 4221, 4246, 4272, 4306, 4326, 4396, 4407, 4458, 4692, 4801, 4818, 4843, 4906, 4962, 4992, 5004, 5102, 5148, 5168, 5172, 5202, 5214, 5334, 5376, 5418, 5462, 5492, 5538, 5604, 5622, 5756, 5784, 6001, 6004, 6008, 6012, 6043, 6045, 6067, 6098, 6264, 6308, 6334, 6368, 6402, 6465, 6542, 6603, 6665, 6714, 6754, 6774, 6834, 6858, 6912, 6998, 7002, 7152, 7242, 7422, 7452, 7476, 7546, 7554, 7564, 7806, 7876, 7896, 7926, 7996, 8003, 8006, 8063, 8154, 8184, 8198, 8289, 8445, 8483, 8598, 8685, 8784, 8948, 9296, 9542, 9672, 9756, 9806, 9836, 9902, 9972, 9976, 10014, 10024, 10062, 10234, 10422, 10548, 10662, 11088, 11118, 11202, 12004, 12504, 12522, 12634, 13308, 13638, 14022, 14448, 14592, 14958, 15114, 15438, 15504, 15648, 15684, 16008, 16038, 16654, 16962, 17742, 18054, 18468, 19398, 20007, 20012, 20112, 20154, 20205, 20289, 20427, 20724, 20784, 21102, 21212, 21272, 21774, 21792, 21962, 22014, 22122, 22164, 22212, 22242, 22401, 22605, 22821, 22824, 22962, 23486, 24086, 24602, 25122, 25386, 26004, 26412, 26574, 26669, 27036, 27252, 27542, 27756, 27912, 28056, 28284, 28896, 29072, 29292, 29402, 29702, 30086, 30098, 30604, 30848, 31234, 32004, 32514, 33336, 33354, 33368, 33726, 34908, 35706, 36124, 36168, 37134, 37204, 37356, 38036, 38184, 39006, 39048

Let's take the last member, 39048, as an example:

• 3 x 9048 - 1 = 27143 is prime
  
  
• 39 x 48 - 1 = 1871 is prime
  
  
• 390 x 48 - 1 = 18719 is prime
  
  
• 3904 x 8 - 1 = 31231 is prime

Semiprime Partitions of Semiprimes

The number associated with my diurnal age today, \( \textbf{28284} \), has a property that grants it membership in OEIS A338902:

A338902: number of integer partitions of the n-th semiprime into semiprimes.

I got Gemini to write a program to find the number of suitable partitions for integers from 1 to 50 displaying the results as both a table and a list (permalink):

Semiprime | Partitions
-----------------------
4         | 1
6         | 1
9         | 1
10        | 2
14        | 3
15        | 2
21        | 4
22        | 7
25        | 7
26        | 10
33        | 17
34        | 25
35        | 21
38        | 34
39        | 34
46        | 73
49        | 87
51        | 103
55        | 149
57        | 176
58        | 206
62        | 281
65        | 344
69        | 479
74        | 725
77        | 881
82        | 1311
85        | 1597
86        | 1742
87        | 1841
91        | 2445
93        | 2808
94        | 3052
95        | 3222
106       | 6784
111       | 9298
115       | 11989
118       | 14533
119       | 15384
121       | 17414
122       | 18581
123       | 19680
129       | 28284
133       | 35862
134       | 38125
141       | 57095
142       | 60582
143       | 64010
145       | 71730
146       | 76016

1, 1, 1, 2, 3, 2, 4, 7, 7, 10, 17, 25, 21, 34, 34, 73, 87, 103, 149, 176, 206, 281, 344, 479, 725, 881, 1311, 1597, 1742, 1841, 2445, 2808, 3052, 3222, 6784, 9298, 11989, 14533, 15384, 17414, 18581, 19680, 28284, 35862, 38125, 57095, 60582, 64010, 71730, 76016

As can be seen, the integer \( \textbf{43} \) has \( \textbf{28284} \) partitions compared to its 63261 unrestricted partitions.

Similar results could be created for primes and triprimes. Here are the result for the primes (permalink). The numbers form OEIS A056768.

Prime   | Partitions
---------------------
2       | 1
3       | 1
5       | 2
7       | 3
11      | 6
13      | 9
17      | 17
19      | 23
23      | 40
29      | 87
31      | 111
37      | 219
41      | 336
43      | 413
47      | 614
53      | 1083
59      | 1850
61      | 2198
67      | 3630
71      | 5007
73      | 5861
79      | 9282
83      | 12488
89      | 19232
97      | 33439
101     | 43709
103     | 49871
107     | 64671
109     | 73506
113     | 94625
127     | 221265
131     | 279516
137     | 394170
139     | 441250
149     | 766262
151     | 853692
157     | 1175344
163     | 1608014
167     | 1975108
173     | 2675925
179     | 3605666
181     | 3977861
191     | 6447003
193     | 7089299
197     | 8559069
199     | 9397474
211     | 16298212
223     | 27810910
227     | 33121140
229     | 36123177

1, 1, 2, 3, 6, 9, 17, 23, 40, 87, 111, 219, 336, 413, 614, 1083, 1850, 2198, 3630, 5007, 5861, 9282, 12488, 19232, 33439, 43709, 49871, 64671, 73506, 94625, 221265, 279516, 394170, 441250, 766262, 853692, 1175344, 1608014, 1975108, 2675925, 3605666, 3977861, 6447003, 7089299, 8559069, 9397474, 16298212, 27810910, 33121140, 36123177

Here are the result for the triprimes (permalink). 

Triprime  | Partitions
-----------------------
8         | 1
12        | 1
18        | 1
20        | 2
27        | 1
28        | 3
30        | 2
42        | 4
44        | 7
45        | 2
50        | 7
52        | 10
63        | 7
66        | 18
68        | 25
70        | 22
75        | 13
76        | 34
78        | 36
92        | 78
98        | 97
99        | 47
102       | 117
105       | 60
110       | 170
114       | 202
116       | 230
117       | 111
124       | 316
125       | 165
130       | 401
138       | 579
147       | 478
148       | 866
153       | 616
154       | 1086
164       | 1652
165       | 1040
170       | 2076
171       | 1350
172       | 2227
174       | 2418
175       | 1566
182       | 3271
186       | 3802
188       | 4115
190       | 4372
195       | 3587
207       | 5708
212       | 9783

1, 1, 1, 2, 1, 3, 2, 4, 7, 2, 7, 10, 7, 18, 25, 22, 13, 34, 36, 78, 97, 47, 117, 60, 170, 202, 230, 111, 316, 165, 401, 579, 478, 866, 616, 1086, 1652, 1040, 2076, 1350, 2227, 2418, 1566, 3271, 3802, 4115, 4372, 3587, 5708, 9783, 13878, 18291, 13983, 22465, 23913, 27489, 29296, 23092, 31441, 32846 (these are first 60 terms).

These results are base-independent and in general what is being done here is to take an integer with a certain property (primeness for example) and express it as a sum of smaller numbers with the same property. The numbere of ways in which this can be done is being counted. Here are some examples:

• 129 = 4 + 4 + 4 + 6 + 111 ... there are 28284 ways in which this can be done
• 97 = 3 + 5 + 89 ... there are 33439 ways in which this can be done
• 212 = 8 + 12 + 18 + 174 ... there are 9783 ways in which this can be done

Rhonda Numbers Revisited

I first reviewed Rhonda Numbers in an eponymous post on the 16th October 2018. Read that post to find out what defines such numbers. Today, upon turning 28280 days old, I was reminded of them again because this number is a member of OEIS A255731:

A255731: Rhonda numbers in sexagesimal number system.

The program that I wrote back then only covered bases from 2 to 36 so I got Gemini to write a new SageMath program that would accommodate every base. Here is what is generated as output when asked to find all Rhonda numbers in base 60 in the range up to one million (permalink):

3348, 3510, 6750, 17430, 18750, 18876, 18944, 19475, 20564, 21312, 26550, 28280, 37230, 38396, 43940, 48042, 77770, 88270, 91224, 97470, 108882, 111403, 120046, 123630, 181996, 182646, 235467, 253460, 260429, 264735, 278675, 289161, 295960, 296055, 306642, 324394, 325593, 337040, 348641, 361221, 377130, 378444, 398274, 411342, 412930, 441048, 444405, 450528, 453470, 458136, 469098, 499533, 503310, 517803, 533731, 534795, 541807, 547515, 566754, 598695, 612374, 612870, 626535, 630410, 656370, 656750, 667491, 670548, 684456, 701765, 703304, 705256, 706275, 709475, 720279, 750225, 757576, 762745, 765245, 809107, 812658, 821106, 877300, 880045, 881454, 915348, 927303, 929830, 930304, 936573, 936675, 967509, 972196, 973549, 984485, 986895, 998430

I also realised that I hadn't included the identification of Rhonda numbers in my daily number analysis and that deficiency has now been remedied. Here the output of the program that I got Gemini to write for the input 29280 (permalink):

28280 : determination of whether it is a Rhonda number in a given base

--- Evaluating 28280 in Base 60 ---

1. Base-60 Digits: [7, 51, 20]

2. Digit Product: 7 * 51 * 20 = 7140

3. Prime Factorization: 2^3 * 5 * 7 * 101

4. Sum of Factors: 2 + 2 + 2 + 5 + 7 + 101 = 119

5. Target Equation (Base * Sum): 60 * 119 = 7140

RESULT: True. 28280 IS a Rhonda number in base 60.

I've set the range of bases to be investigate to 100,000 so that even a number like 889200 with is a Rhonda number in TEN different bases has an accurate output:

--- Evaluating 889200 in Base 1512 ---

1. Base-1512 Digits: [588, 144]

2. Digit Product: 588 * 144 = 84672

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 1512 * 56 = 84672

RESULT: True. 889200 IS a Rhonda number in base 1512.

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

--- Evaluating 889200 in Base 2760 ---

1. Base-2760 Digits: [322, 480]

2. Digit Product: 322 * 480 = 154560

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 2760 * 56 = 154560

RESULT: True. 889200 IS a Rhonda number in base 2760.

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

--- Evaluating 889200 in Base 5160 ---

1. Base-5160 Digits: [172, 1680]

2. Digit Product: 172 * 1680 = 288960

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 5160 * 56 = 288960

RESULT: True. 889200 IS a Rhonda number in base 5160.

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

--- Evaluating 889200 in Base 7904 ---

1. Base-7904 Digits: [112, 3952]

2. Digit Product: 112 * 3952 = 442624

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 7904 * 56 = 442624

RESULT: True. 889200 IS a Rhonda number in base 7904.

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

--- Evaluating 889200 in Base 9400 ---

1. Base-9400 Digits: [94, 5600]

2. Digit Product: 94 * 5600 = 526400

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 9400 * 56 = 526400

RESULT: True. 889200 IS a Rhonda number in base 9400.

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

--- Evaluating 889200 in Base 10032 ---

1. Base-10032 Digits: [88, 6384]

2. Digit Product: 88 * 6384 = 561792

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 10032 * 56 = 561792

RESULT: True. 889200 IS a Rhonda number in base 10032.

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

--- Evaluating 889200 in Base 11440 ---

1. Base-11440 Digits: [77, 8320]

2. Digit Product: 77 * 8320 = 640640

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 11440 * 56 = 640640

RESULT: True. 889200 IS a Rhonda number in base 11440.

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

--- Evaluating 889200 in Base 12920 ---

1. Base-12920 Digits: [68, 10640]

2. Digit Product: 68 * 10640 = 723520

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 12920 * 56 = 723520

RESULT: True. 889200 IS a Rhonda number in base 12920.

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

--- Evaluating 889200 in Base 14136 ---

1. Base-14136 Digits: [62, 12768]

2. Digit Product: 62 * 12768 = 791616

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 14136 * 56 = 791616

RESULT: True. 889200 IS a Rhonda number in base 14136.

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

--- Evaluating 889200 in Base 15080 ---

1. Base-15080 Digits: [58, 14560]

2. Digit Product: 58 * 14560 = 844480

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 15080 * 56 = 844480

RESULT: True. 889200 IS a Rhonda number in base 15080.

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

28277: A Prime To Be Proud Of

Until today my diurnal age has suffered a drought of prime numbers. Prior to today (3rd September 2026) when I turned 28277 days old, the last prime (28229) occurred on July 17th. This marked a gap of 48 days between successive primes. This is not a record gap but it is impressive. Figure 1 shows the successive record gaps between primes.

Figure 1

I'll enumerate some of 28277's most interesting properties:

PROPERTY 1:

It forms a twin prime with 28279 but it also marks the beginning of gaps of 2, 4, 6, 8 and 10 between successive primes. The progression of primes is thus 28277, 28279, 28283, 28289, 28297, 28307. Such an occurrence is not common and membership is restricted to only three numbers (13901, 21557, 28277) in the range up to 40000. These and subsequent numbers constitute OEIS A190817.

PROPERTY 2

28277 is what is called a "good" prime and I posted about this type of prime in my blog post titled The Good Prime on the 12th of March 2025. As I explained there:

A prime \(p_n\) is said to be \( \textbf{good} \) if \(p_n^{^\textbf{2}}>p_{n-i } \cdot p_{n+i} \) for all \( 1 \leq i < n \).

The good primes from 28277 to 40000 are: 

28277, 28387, 28403, 28493, 28537, 28571, 28591, 28597, 29833, 29983, 30011, 30059, 30089, 30491, 30631, 30637, 30671, 30757, 30803, 31121, 31139, 31147, 31957, 32027, 32051, 32057, 32297, 32969, 33287, 33311, 33329, 34123, 35729, 35747, 35797, 35801, 35831, 35951, 35963, 36433, 36451, 36467, 36523, 36527, 36671, 38113, 38149, 38167, 38177, 38543, 38557, 38593, 38651, 38669, 39079, 39089

PROPERTY 3

28277 has what might be called an "internal prime". Strip away the first and last digits and what remains is 827, a prime number. I discuss these types of numbers in my post titled Numbers Within Numbers from the 21st June 2026. Primes with this property form OEIS A069686:

 A069686: primes whose internal digits form a prime.

From 28277 to 40000, the members of the sequence are:

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

PROPERTY 4

28277 gives prime 31931717 when digits become indices of prime numbers. Here we have:

• \(2 \rightarrow p_2=3\)
• \(8 \rightarrow p_8=19\)
• \(7 \rightarrow p_7=17\)

The primes of this sort from 28277 to 40000 are (permalink):

28277, 28289, 28319, 28429, 28807, 28979, 29137, 29347, 29399, 29717, 29819, 29837, 29917, 30059, 30089, 30169, 30187, 30367, 30467, 30469, 30497, 30509, 30689, 30697, 30707, 30727, 31139, 31159, 31177, 31247, 31277, 31337, 31469, 31489, 31687, 31847, 31849, 32099, 32309, 32569, 32717, 32957, 32987, 33049, 33469, 33577, 33629, 33797, 33809, 33889, 33937, 34019, 34129, 34259, 34297, 34327, 34439, 34469, 34607, 34649, 34807, 34819, 35027, 35159, 35407, 35597, 36037, 36107, 36109, 36467, 36469, 36587, 36637, 36809, 36857, 37087, 37139, 37199, 37277, 37337, 37579, 37889, 38189, 38299, 38327, 38459, 38609, 38639, 38699, 38747, 39047, 39119, 39239, 39397, 39667, 39779

PROPERTY 5

28277 is the average of a prime and its emirp in two different ways. The two ways are:

1. \( \dfrac{18773 + 37781}{2}= 28277\)
2. \( \dfrac{19763 + 36791}{2} = 28277\)

I discuss these sorts of primes in my blog post Prime Emirp Pair Averages of 12th May 2023. Such primes form OEIS A178587:

A178587

Primes that are the average of the members of more than one emirp pair.

These primes are few and far between with the initial members being:

14741, 22727, 23327, 24547, 25447, 27067, 28277, 42929, 63541, 65761, 85453, 1217171, 1221221, 1227271, 1243421, 1245421, 1246471, 1250521, 1253521, 1257521, 1261571, 1271671, 1283771, 1327231, 1335331, 1338331, 1339381 

PROPERTY 6

28277 has the following trajectory under the Primes(+) and Non_Primes(-) algorithm: 

\(28277 \rightarrow 28287 \rightarrow 28282 \rightarrow 28272 \rightarrow 28277\)

It is thus a vortical and part of the vortex beginning and ending with 28277. This vortex has 36 captives.

Let Us Count the Ways

I was curious as to how to count the number of ways that a given number can be represented in Loeschian form based on its prime factorisation. I knew there was such a method for counting how many ways a number can be represented as a sum of two squares. I got Gemini to investigate, compare and summarise the results. Notice the use of the ceiling function when determining the number of essentially distinct pairs. Remember also that the invalid primes become valid if raised to an even power but they don't contribute to the count. The \(b_{\textit{i}}\) in the table refers to the exponents of the generating primes.

The algebraic frameworks for expressing integers as \(x^2 + xy + y^2\) and \(x^2 + y^2\) mirror each other directly through their respective complex integer rings.

Characteristic	Loeschian Form \(x^2 + xy + y^2\)	Sum of Two Squares \(x^2 + y^2\)
Complex Ring	Eisenstein Integers \(\mathbb{Z}[e^{2\pi i/3}]\)	Gaussian Integers \(\mathbb{Z}[i]\)
Invalid Primes (Require Even Exponents)	\(p \equiv 2 \pmod 3\)	\(p \equiv 3 \pmod 4\)
Generating Primes (Exponents \(b_i\))	\(p \equiv 1 \pmod 3\)	\(p \equiv 1 \pmod 4\)
Neutral Prime	\(p = 3\)	\(p = 2\)
Unit Symmetries (Multiplier)	6	4
Total Ordered Pairs \((x,y)\)	\(6 \prod (b_i + 1)\)	\(4 \prod (b_i + 1)\)
Essentially Distinct Pairs \(\{x,y\}\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)

Factorization Conditions

Both systems demand that primes which cannot be represented natively by the quadratic form must be "squared away" with even exponents in the prime factorization. For Gaussian integers, primes congruent to \(3 \pmod 4\) (such as 3, 7, 11) cannot form sums of two squares. For Eisenstein integers, primes congruent to \(2 \pmod 3\) (such as 2, 5, 11) cannot be Loeschian. If any of these restricted primes have an odd exponent, the integer cannot be represented in that form, and the total count is zero.

Symmetry and Distinct Ways

The difference in the total number of ordered coordinate pairs stems entirely from the geometry of the complex rings. Gaussian integers form a square lattice with 4 units (\(\pm 1, \pm i\)), leading to a rotational symmetry multiplier of 4. Eisenstein integers form a hexagonal lattice with 6 units (\(\pm 1, \pm e^{2\pi i/3}, \pm e^{4\pi i/3}\)), leading to a multiplier of 6.

Once these geometric symmetries are factored out by dividing by the respective unit count, both systems calculate the number of essentially distinct, non-negative integer pairs using the exact same ceiling function formula. The generating primes dictate this final count, while the unique neutral prime in each system scales the base integer without creating new algebraic variations.