Some Rare Sextuplets and Septuplets

In terms of the numbers associated with my diurnal age, I recently concluded a run of six numbers (a sextuplet) with the shared property that they could all be written as a product of three, not necessarily distinct, prime factors. These were the numbers together with their factorisations:

• 27290 = 2 * 5 * 2729

• 27291 = 3 * 11 * 827

• 27292 = 2^2 * 6823

• 27293 = 7^2 * 557

• 27294 = 2 * 3 * 4549

• 27295 = 5 * 53 * 103

Four of the members of this sextuplet are sphenic but they are not consecutive because, counting from 4, every fourth number is subsequently divisible by 4. As 27290 marks the beginning of this run of numbers, it qualifies for membership in OEIS A045942:

A045942

Numbers \(m\) such that the factorisations of \(m \dots m+5\) have the same number of primes (including multiplicities).

Such numbers are few and far between. The initial members of the sequence are:

2522, 4921, 18241, 25553, 27290, 40313, 90834, 95513, 98282, 98705, 117002, 120962, 136073, 136865, 148682, 153794, 181441, 181554, 185825, 204323, 211673, 211674, 212401, 215034, 216361, 231002, 231665, 234641, 236041, 236634, 266282, 281402, 284344, 285410

As can be seen, I'm unlikely to experience the next number (of days old) which is 40313 by which time I'd be 110 years old. Some of the numbers in this sequence mark the start of a run of seven numbers, a septuplet, and this can be seen in by the two adjacent numbers in the list above (211673 and 211674). 211673 and similar numbers constitute OEIS A123103:

A123103

Numbers \(m\) such that the factorisations of \(m \dots m+6\) have the same number of primes (including multiplicities).

The initial members are:

211673, 298433, 355923, 381353, 460801, 506521, 540292, 568729, 690593, 705953, 737633, 741305, 921529, 1056529, 1088521, 1105553, 1141985, 1187121, 1362313, 1721522, 1811704, 1828070, 2016721, 2270633, 2369809, 2535721, 2590985

Let's look at the septuplet beginning with 211673:

• 211673 = 7 * 11 * 2749

• 211674 = 2 * 3 * 35279

• 211675 = 5^2 * 8467

• 211676 = 2^2 * 52919

• 211677 = 3 * 37 * 1907

• 211678 = 2 * 109 * 971

• 211679 = 13 * 19 * 857

There are longer runs but the numbers then become very large: A358017 (\(k=8\)), A358018 (\(k=9\)), A358019 (\(k=10\)).

ADDENDUM: March 6th 2024

Today I created a post that more or less reproduced what I've covered here. I had to delete it because it was largely redundant. Now that I have so many posts it's important that I check first before I create posts. The hashtags become very important in this context.