A215150 - OEIS

%I #18 Apr 03 2023 10:36:13

%S 13981,18705,23001,55245,63973,72885,75361,107185,126217,129921,

%T 137149,157641,158369,172081,176149,188461,215265,266305,272251,

%U 276013,278545,285541,294409,348161,387731,423793,464185,488881,493697,617093,625921,743665,748657,825265

%N Pseudoprimes divisible by a smaller pseudoprime.

%C Here pseudoprime means a Fermat base-2 pseudoprime (a member of A001567).

%C Pseudoprimes by which the numbers from sequence are divisible: 341, 645, 561, 1905, 1729, 645, 341, 1105, 1387 and 1729, 341, 2047, 561, 5461, 2821, 1387, 1729, 1905, 1105, 2047, 1387, 2465 and 3277, 4681, 2701 and 4033 and 7957, 341, 5461, 4369, 5461, 2701, 4369, 5461, 10261, 1105, 1729, 1387 and 11305.

%C A pseudoprime can be divisible by more than one pseudoprime: e.g. 126217, 278545, 294409, 825265.

%H Arkadiusz Wesolowski, <a href="/A215150/b215150.txt">Table of n, a(n) for n = 1..1000</a>

%H G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/25271.html">Prime Curios! 13981</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%e Since 2^13980 = 1 mod 13981 and 13981 = 11 * 31 * 41, 13981 is a pseudoprime, and it is divisible by 341, a smaller pseudoprime. 13981 is therefore in the sequence.

%e The pseudoprimes 75361, 129921, 348161, etc., are also divisible by 341.

%t lst1 = {}; lst2 = {}; r = 10^6; Do[If[! PrimeQ[n] && PowerMod[2, n - 1, n] == 1, AppendTo[lst1, n]], {n, 1, r, 2}]; l = Length[lst1]; Do[p = lst1[[a]]; b = 1; While[True, t = lst1[[b]]; If[p < 3*t, Break[]]; If[Divisible[p, t], AppendTo[lst2, p]; Break[]]; b++], {a, 2, l}]; lst2

%Y Cf. A001567, A214305.

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Aug 04 2012