This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A215343 #15 Feb 16 2025 08:33:18 %S A215343 831405,5977153,15913261,21474181,38171953,126619741,210565981, %T A215343 224073865,327718401,377616421,390922741,558097345,699735345, %U A215343 700932961,1327232481,1999743661,4996150741,8523152641,11358485281,13999580785,15613830541,17657245081,20442723301 %N A215343 Fermat pseudoprimes to base 2 that can be written as 2*p^2 - p, where p is also a Fermat pseudoprime to base 2. %C A215343 Fermat pseudoprimes are listed in A001567. %C A215343 The correspondent p for the numbers from the sequence above: 645, 1729, 2821, 3277, 4369, 7957, 10261, 10585, 12801, 13741, 13981, 16705, 18705, 25761, 31621, 49981, 65281, 75361, 83665, 88357, 93961, 101101. %C A215343 Note that for 22 of the first 80 Poulet numbers, we obtained through this formula another Poulet number! %C A215343 The formula could be generalized this way: Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0, and p is another Poulet number. %C A215343 For n = 1, that formula becomes the formula set out for the sequence above. %C A215343 For n = 2, that formula becomes 3*p^2 - 2*p, from which the Poulet numbers 348161 (for p = 341) and 1246785 (for p = 645) were obtained. %C A215343 For n = 3, that formula becomes 4*p^2 - 3*p, from which the Poulet number 119273701 (for p = 5461) was obtained. %C A215343 For n = 4, that formula becomes 5*p^2 - 4*p, from which the Poulet numbers 2077545 (for p = 645) and 9613297 (for p = 1387) were obtained. %C A215343 Conjecture: there are infinitely many Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0, and p is another Poulet number. %C A215343 Finally, considering, e.g., that for the Poulet number 645, Poulet numbers were obtained for n = 1, 2, 4 (i.e., 831405, 1246785, 2077545), yet another conjecture: For any Poulet number p, there are infinitely many Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0. %H A215343 Charles R Greathouse IV, <a href="/A215343/b215343.txt">Table of n, a(n) for n = 1..10000</a> %H A215343 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a> %Y A215343 Cf. A001567, A213812. %K A215343 nonn %O A215343 1,1 %A A215343 _Marius Coman_, Aug 08 2012 %E A215343 Edited by _Jon E. Schoenfield_, Dec 12 2013 %E A215343 a(14) inserted by _Charles R Greathouse IV_, Jul 07 2017