Mathematical Meanderings: P-Composite Numbers

Let's create a class of positive integers called \(p\)-composite integers where \(p\) stands for any prime number. For a number to be so-called, it must satisfy the following criteria:

it must be composite
it's sum of digits must equal \(p\)
\(p\) must be a divisor of the number

Let's take 1679 as an example. It's sum of digits is 23 and 23 divides it to give 73. Thus 1679 can be termed a 23-composite number. The number of 23-composite numbers in the range up to 40000 is 94. The numbers are:

23-composite

1679, 1886, 3749, 3956, 4577, 4784, 4991, 5198, 5819, 6647, 6854, 7268, 7475, 7682, 8096, 8717, 8924, 9338, 9545, 9752, 10787, 10994, 12857, 13478, 13685, 13892, 14099, 14927, 15548, 15755, 15962, 16169, 16376, 16583, 16790, 17618, 17825, 18239, 18446, 18653, 18860, 19067, 19274, 19481, 21758, 21965, 22379, 22586, 22793, 23828, 24449, 24656, 24863, 25277, 25484, 25691, 26519, 26726, 26933, 27347, 27554, 27761, 28175, 28382, 29417, 29624, 29831, 30659, 30866, 31487, 31694, 32729, 32936, 33557, 33764, 33971, 34178, 34385, 34592, 35627, 35834, 36248, 36455, 36662, 37076, 37283, 37490, 37904, 38318, 38525, 38732, 39146, 39353, 39560

Let's work our way up the primes from 2.

2-composite

20, 110, 200, 1010, 1100, 2000, 10010, 10100, 11000, 20000

3-composite

12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000

5-composite

50, 140, 230, 320, 410, 500, 1040, 1130, 1220, 1310, 1400, 2030, 2120, 2210, 2300, 3020, 3110, 3200, 4010, 4100, 5000, 10040, 10130, 10220, 10310, 10400, 11030, 11120, 11210, 11300, 12020, 12110, 12200, 13010, 13100, 14000, 20030, 20120, 20210, 20300, 21020, 21110, 21200, 22010, 22100, 23000, 30020, 30110, 30200, 31010, 31100, 32000

7-composite

70, 133, 322, 511, 700, 1015, 1141, 1204, 1330, 2023, 2212, 2401, 3031, 3220, 4102, 5110, 7000, 10024, 10150, 10213, 10402, 11032, 11221, 11410, 12040, 12103, 13111, 13300, 15001, 20041, 20104, 20230, 21112, 21301, 22120, 23002, 24010, 30121, 30310, 31003, 32011, 32200

11-composite

209, 308, 407, 506, 605, 704, 803, 902, 2090, 3080, 4070, 5060, 6050, 7040, 8030, 9020, 10109, 10208, 10307, 10406, 10505, 10604, 10703, 10802, 10901, 20009, 20108, 20207, 20306, 20405, 20504, 20603, 20702, 20801, 20900, 30008, 30107, 30206, 30305, 30404, 30503, 30602, 30701, 30800

13-composite

247, 364, 481, 715, 832, 1066, 1183, 1417, 1534, 1651, 2119, 2236, 2353, 2470, 2704, 2821, 3055, 3172, 3406, 3523, 3640, 4108, 4225, 4342, 4810, 5044, 5161, 5512, 6214, 6331, 7033, 7150, 7501, 8203, 8320, 9022, 10075, 10192, 10309, 10426, 10543, 10660, 11128, 11245, 11362, 11713, 11830, 12064, 12181, 12415, 12532, 13117, 13234, 13351, 13702, 14053, 14170, 14404, 14521, 15106, 15223, 15340, 16042, 16510, 17212, 18031, 19201, 20137, 20254, 20371, 20605, 20722, 21073, 21190, 21307, 21424, 21541, 22009, 22126, 22243, 22360, 22711, 23062, 23413, 23530, 24115, 24232, 24700, 25051, 25402, 26104, 26221, 27040, 28210, 30082, 30316, 30433, 30550, 30901, 31018, 31135, 31252, 31603, 31720, 32071, 32305, 32422, 33007, 33124, 33241, 34060, 34411, 35113, 35230, 36400, 37102

17-composite

476, 629, 782, 935, 1088, 1394, 1547, 1853, 2159, 2465, 2618, 2771, 2924, 3077, 3383, 3536, 3842, 4148, 4454, 4607, 4760, 4913, 5066, 5219, 5372, 5525, 5831, 6137, 6290, 6443, 6902, 7055, 7208, 7361, 7514, 7820, 8126, 8432, 9044, 9350, 9503, 10268, 10574, 10727, 10880, 11186, 11339, 11492, 11645, 11951, 12257, 12563, 12716, 13175, 13328, 13481, 13634, 13940, 14093, 14246, 14552, 14705, 15164, 15317, 15470, 15623, 16082, 16235, 16541, 17153, 17306, 17612, 18071, 18224, 18530, 19142, 19601, 20366, 20519, 20672, 20825, 21284, 21437, 21590, 21743, 22049, 22355, 22508, 22661, 22814, 23273, 23426, 23732, 24038, 24191, 24344, 24650, 24803, 25109, 25262, 25415, 25721, 26027, 26180, 26333, 27251, 27404, 27710, 28016, 28322, 29240, 30158, 30464, 30617, 30770, 30923, 31076, 31229, 31382, 31535, 31841, 32147, 32453, 32606, 32912, 33065, 33218, 33371, 33524, 33830, 34136, 34442, 34901, 35054, 35207, 35360, 35513, 36125, 36431, 37043, 37502, 38114, 38420, 39032

19-composite

874, 1387, 1558, 1729, 2584, 2755, 2926, 3097, 3268, 3439, 3781, 3952, 4294, 4465, 4636, 4807, 5149, 5491, 5662, 5833, 6175, 6346, 6517, 7372, 7543, 7714, 8056, 8227, 8740, 8911, 9082, 9253, 9424, 10279, 10792, 10963, 11476, 11647, 11818, 12673, 12844, 13186, 13357, 13528, 13870, 14383, 14554, 14725, 15067, 15238, 15409, 15580, 15751, 15922, 16093, 16264, 16435, 16606, 17119, 17290, 17461, 17632, 17803, 18145, 18316, 19171, 19342, 19513, 20197, 20368, 20539, 20881, 21394, 21565, 21736, 21907, 22078, 22249, 22591, 22762, 22933, 23275, 23446, 23617, 24472, 24643, 24814, 25156, 25327, 25840, 26182, 26353, 26524, 27037, 27208, 27550, 27721, 28063, 28234, 28405, 29260, 29431, 29602, 30286, 30457, 30628, 30970, 31483, 31654, 31825, 32167, 32338, 32509, 32680, 32851, 33193, 33364, 33535, 33706, 34048, 34219, 34390, 34561, 34732, 34903, 35074, 35245, 35416, 36271, 36442, 36613, 37126, 37810, 38152, 38323, 39007, 39520

29-composite

4988, 7598, 7859, 9686, 9947, 15689, 16994, 17777, 18299, 19865, 22997, 25868, 27695, 27956, 28478, 28739, 29783, 33698, 33959, 35786, 36569, 37874, 38396, 38657, 38918, 39179, 39962

31-composite

8959, 9796, 17887, 25699, 25978, 28489, 28768, 29884, 36859, 37696, 37975, 39649, 39928

37-composite

37999, 38998, 39997

41-composite