Sphenic Numbers and Palindromes

In this post, I want to look at some connections between between sphenic numbers and palindromes. The most obvious link is to those sphenic numbers that are palindromic. In the range up to 100,000, there are 229 palindromic sphenic numbers. The list is shown below.

66, 222, 282, 434, 474, 494, 555, 595, 606, 646, 777, 969, 1001, 1221, 1551, 1771, 2222, 2882, 3333, 3553, 4334, 4994, 5335, 5555, 5665, 5885, 5995, 6226, 6446, 6886, 7337, 7557, 7667, 7777, 7887, 8338, 8558, 8998, 9339, 9669, 9779, 9889, 11211, 11811, 12121, 12621, 12921, 13731, 14241, 14541, 15051, 15951, 16261, 16761, 17171, 18381, 18681, 19491, 19591, 19691, 20002, 20702, 20802, 20902, 22222, 22922, 24042, 24342, 24542, 24742, 24942, 26062, 26162, 26462, 28082, 28282, 28382, 28582, 28882, 28982, 30003, 30503, 31413, 31913, 32123, 32223, 32523, 32623, 32923, 33333, 33733, 34143, 34743, 35553, 37373, 37973, 38283, 38883, 39093, 39193, 39693, 39893, 41114, 41214, 41914, 43334, 43934, 45154, 45354, 45854, 47174, 47274, 47474, 49594, 49994, 50005, 50105, 50205, 50405, 50605, 51815, 51915, 53635, 53735, 54245, 54345, 54645, 54845, 55155, 55255, 55455, 55555, 56165, 56465, 56665, 58785, 58985, 59095, 59395, 59495, 60106, 60706, 60906, 62326, 62626, 62726, 62926, 64046, 64146, 64546, 64946, 66266, 66466, 66566, 66866, 66966, 68086, 68186, 68386, 68686, 68786, 68986, 70007, 70807, 70907, 71517, 71617, 71817, 72127, 72427, 72527, 72627, 73337, 74847, 75057, 75257, 75757, 75957, 76067, 76467, 76867, 77577, 77777, 78987, 79097, 79597, 79797, 81318, 81818, 83138, 83238, 83738, 85058, 85258, 85458, 85558, 85758, 87378, 87478, 87878, 89198, 89498, 89798, 89898, 89998, 91119, 91419, 91819, 92229, 92729, 92929, 93439, 93939, 94449, 94549, 95259, 95459, 95559, 96069, 97179, 97279, 97679, 97779, 98589, 99199, 99399, 99499, 99699, 99799

How many of these numbers have prime factors that sum to a palindrome? Well, as it turns out, only 21. These are shown below along with the factorisations and prime factor sums:

• 11211 = 3 * 37 * 101 --> 141
• 11811 = 3 * 31 * 127 --> 161
• 14241 = 3 * 47 * 101 --> 151
• 14541 = 3 * 37 * 131 --> 171
• 16261 = 7 * 23 * 101 --> 131
• 16761 = 3 * 37 * 151 --> 191
• 19591 = 11 * 13 * 137 --> 161
• 20002 = 2 * 73 * 137 --> 212
• 22922 = 2 * 73 * 157 --> 232
• 26062 = 2 * 83 * 157 --> 242
• 26162 = 2 * 103 * 127 --> 232
• 28582 = 2 * 31 * 461 --> 494
• 31413 = 3 * 37 * 283 --> 323
• 31913 = 7 * 47 * 97 --> 151
• 32123 = 7 * 13 * 353 --> 373
• 32523 = 3 * 37 * 293 --> 333
• 34743 = 3 * 37 * 313 --> 353
• 50605 = 5 * 29 * 349 --> 383
• 54245 = 5 * 19 * 571 --> 595
• 58985 = 5 * 47 * 251 --> 303
• 68686 = 2 * 61 * 563 --> 626

It's interesting how there are no palindromic sphenic numbers after 68686 in the range up to one hundred thousand. I decided to extend the range to one million and I found that there are surprisingly few in the range between one hundred thousand and one million. They are as follows (permalink):

• 122221 = 11 * 41 * 271 --> 323
• 518815 = 5 * 11 * 9433 --> 9449
• 713317 = 11 * 19 * 3413 --> 3443
• 751157 = 11 * 23 * 2969 --> 3003
• 760067 = 7 * 11 * 9871 --> 9889
• 961169 = 11 * 59 * 1481 --> 1551

We can relax our palindromic criteria and ask what sphenic numbers have prime factor sums that are palindromic. Here we are relaxing the condition that the sphenic number itself must be palindromic. In this case, there are 1403 sphenic numbers that satisfy in the range up to one hundred thousand. If we extend the range to one million, only 7172 satisfy which is about half the number expected if the previous rate remained relatively constant (permalink). 

What got me thinking about this was the number associated with my diurnal age for May 2nd 2023. This number is 27057 and its prime factors (3, 29 and 311) sum to 343. It's interesting to count the number of times that these palindromic sums occur. For instance, in the range up to one hundred thousand, the palindromic sums of 131 and 161 occur 51 times each. This is shown in Figure 1 where the number of times all sums occur is shown.

Figure 1: permalink