Neglected Numbers

Sometimes it's easy to give up on numbers. Take 26271 for example. This number represents my diurnal age today and when I searched in the OEIS nothing of interest showed up. Even a search using OEIS + 26271, which often throws up OEIS b-files containing the number, didn't reveal anything interesting. 

Figure 1: link

In such situations I fall back on the Sequence Database: a database with 2402879 machine generated integer and decimal sequences. I discovered this site in June of 2020. Unfortunately, this didn't turn up any sequences of interest either but then I remembered another site that lists sequences (see Figure 1). 

It was here that I found something of substance. See Figure 2.

Figure 2

26271 belongs to a sequence of numbers \(n\) with the property that \(n\) is 5-almost prime and \(n+1 \) is 6-almost prime. In the case of 26271, it factors to \(3^3 \times 7 \times 139 \) and 26272 factors to \(2^5 \times 821\). The example of 728, the first member of the sequence, is given in Figure 1.

It's easy enough to generate this sequence in SageMath, using the algorithm shown in Figure 3:

Figure 3: permalink

This algorithm can be easily modified to search for other patterns involving \(k\)-almost primes where \(k \geq 2\). For example, what numbers \(n\) are there, up to 26271, with the property that \(n\) is 4-almost prime and \(n+1 \) is 5-almost prime? These numbers turn out to be quite numerous and I won't list them all here but notice how 26270 makes an appearance:

495, 975, 1071, 1287, 1484,  ... , 26075, 26103, 26195, 26215, 26270

From this we can see that 26271 is the middle term in a triplet of numbers with the property that: 

• \(26270 = 2 \times 5 \times 37 \times 71\) and is 4-almost prime
• \(26271 = 3^3 \times 7 \times 139 \) and is 5-almost prime
• \(26272 = 2^5 \times 821\) and is 6-almost prime

This condition could be a good candidate for admission to the OEIS. It could be framed as follows:

Numbers \(n\) such that \(n-1\) is 4-almost prime, \(n\) is 5-almost prime and \(n+1\) is 6-almost prime.

Up to 100,000 the members of this sequence are:

11151, 13455, 23375, 26271, 31311, 33776, 36125, 40375, 45495, 46375, 48411, 49049, 49167, 61335, 63125, 74151, 77895, 78111, 78351, 80271, 82575, 83511, 84591, 86031, 87375, 88749, 90207

I checked and this sequence is not currently in the OEIS so I'll put it forward as a candidate and report back here when it's approved.

Reporting back, I'm happy to report that the sequence was approved, although one of the arbiters took exception to my use of the term \(n\)-almost prime and suggested that it be replaced by composite with \(n\) prime factors. I followed this suggestion (always a good idea to keep the arbiters on side) and the result is OEIS A342246:

A342246

Numbers \(n\) such that \(n-1\), \(n\) and \(n+1\) are all composite with four, five and six (not necessarily distinct) prime factors respectively

Before I discovered the 5-almost prime and 6-almost prime link on the Italian site, I also made a discovery of my own. 26271 is a member of a sequence of numbers such that the sum of its digits is equal to the sum of the digits of its totient. Here the digits of 26271 add to 18 and the digits of its totient 14904 also add to 18. Up to 26271, about 7.61% of numbers have this property. The members of this sequence, up to 1000, are as follows:

1, 21, 27, 34, 54, 63, 81, 106, 108, 117, 126, 129, 135, 136, 142, 147, 156, 162, 171, 178, 195, 205, 212, 214, 216, 234, 237, 243, 252, 270, 272, 291, 315, 324, 333, 342, 351, 356, 358, 394, 402, 405, 424, 432, 441, 459, 493, 502, 504, 513, 538, 540, 544, 565, 585, 624, 630, 663, 702, 712, 714, 716, 718, 723, 729, 745, 804, 810, 831, 834, 835, 840, 864, 873, 918, 932, 934, 936, 943, 981

This discovery also prompted me to investigate the sum of the digits of the sum of the divisors. The percentage of numbers whose sum of digits equals the sum of the digits of the sum of its divisors is 4.59%. The members of this sequence, up to 1000, are as follows:

1, 15, 24, 64, 69, 78, 90, 114, 133, 147, 153, 186, 198, 258, 270, 276, 288, 306, 339, 360, 366, 393, 429, 474, 492, 495, 507, 522, 582, 588, 609, 618, 627, 639, 708, 717, 738, 762, 763, 801, 817, 834, 846, 871, 906, 933, 960, 978, 990

The percentage of numbers whose sum of digits equals both the sum of the digits of the sum of its divisors AND the sum of the digits of its totient is 0.727%. The members of this sequence, up to 10000, are as follows:

1, 147, 270, 834, 1158, 1374, 1377, 2028, 2214, 2436, 2454, 2538, 2592, 2646, 2754, 2862, 2907, 3339, 3726, 4293, 4320, 4371, 4614, 5049, 5262, 5319, 5472, 5508, 5607, 5670, 5751, 5802, 6234, 6291, 6426, 6570, 6972, 7155, 7209, 7434, 7560, 7605, 7614, 8190, 8934, 9126, 9270, 9315, 9333, 9342, 9720, 9744, 9810

This last sequence might be an interesting one for admission to the OEIS (as it's not currently in there) but I'll wait until my current submission is approved. This is again easy to generate in SageMath as Figure 4 shows:

Figure 4: permalink

In conclusion, it can be said that 26271 is an Ulam number, being the sum of two previous Ulam numbers, namely 47 and 26224. It is also a happy number since repeated sums of squares of digits lead to 1. However, as we've seen, it also has other interesting properties that a little research uncovered.