A327789 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).

4369, 1387, 341, 3277, 2047, 8321, 31621, 104653, 280601, 13747, 2081713, 88357, 8902741, 741751, 665333, 680627, 2008597, 1252697, 3235699, 1293337, 513629, 8095447, 83333, 2284453, 604117, 191981609, 1530787, 13747361, 3568661, 769757, 6973063, 275887, 12854437, 16705021

Offset

1,1

Comments

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.

Links

Daniel Suteu, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Poulet Number

Example

a(1) = 4369 = (2*2*2*2 + 1)(2*2*2*2*2*2*2*2 + 1).
a(2) = 1387 = (2*3*3 + 1)(2*2*2*3*3 + 1).
a(3) = 341 = (2*5 + 1)(2*3*5 + 1).
a(4) = 3277 = (2*2*7 + 1)(2*2*2*2*7 + 1).
a(5) = 2047 = (2*11 + 1)(2*2*2*11 + 1).

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A347603 A217345 A140937 * A043598 A043843 A043857

Adjacent sequences: A327786 A327787 A327788 * A327790 A327791 A327792

Keyword

nonn

Author

Daniel Suteu, Sep 25 2019

Status

approved