A327789 - OEIS

%I #11 Oct 10 2019 04:24:00

%S 4369,1387,341,3277,2047,8321,31621,104653,280601,13747,2081713,88357,

%T 8902741,741751,665333,680627,2008597,1252697,3235699,1293337,513629,

%U 8095447,83333,2284453,604117,191981609,1530787,13747361,3568661,769757,6973063,275887,12854437,16705021

%N a(n) is the smallest Fermat pseudoprime to base 2 such that gpf(p-1) = prime(n) for all prime factors p of a(n).

%C Equivalently, a(n) is the smallest composite number k such that 2^(k-1) == 1 (mod k) and gpf(p-1) = prime(n) for all prime factors p of k.

%H Daniel Suteu, <a href="/A327789/b327789.txt">Table of n, a(n) for n = 1..10000</a>

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

%e a(1) = 4369 = (2*2*2*2 + 1)(2*2*2*2*2*2*2*2 + 1).

%e a(2) = 1387 = (2*3*3 + 1)(2*2*2*3*3 + 1).

%e a(3) = 341 = (2*5 + 1)(2*3*5 + 1).

%e a(4) = 3277 = (2*2*7 + 1)(2*2*2*2*7 + 1).

%e a(5) = 2047 = (2*11 + 1)(2*2*2*11 + 1).

%t pspQ[n_] := CompositeQ[n] && PowerMod[2, (n - 1), n] == 1; gpf[n_] := FactorInteger[n][[-1, 1]]; g[n_] := If[Length[(u = Union[gpf /@ (FactorInteger[n][[;; , 1]] - 1)])] == 1, u[[1]], 1]; m = 10; c = 0; k = 0; v = Table[0, {m}]; While[c < m, k++ If[! pspQ[k], Continue[]]; If[(p = g[k]) > 1, i = PrimePi[p]; If[i <= m && v[[i]] == 0, c++; v[[i]] = k]]]; v (* _Amiram Eldar_, Oct 08 2019 *)

%o (Perl) use ntheory ":all"; sub a { my $p = nth_prime(shift); for(my $k = 4; ; ++$k) { return $k if (is_pseudoprime($k,2) and !is_prime($k) and vecall { (factor($_-1))[-1] == $p } factor($k)) } }

%o for my $n (1..25) { print "a($n) = ", a($n), "\n" }

%Y Cf. A001567 (Fermat pseudoprimes to base 2), A006530 (gpf).

%K nonn

%O 1,1

%A _Daniel Suteu_, Sep 25 2019