A178705 Odd composite numbers q such that there exists a, 2<=a<=q-2, such that a^d == 1 mod q where d = A000265(q-1). Thus q is a strong pseudoprime in base a.

49, 91, 121, 133, 169, 175, 217, 231, 247, 259, 301, 325, 341, 343, 361, 385, 403, 427, 435, 451, 469, 475, 481, 511, 529, 553, 559, 561, 589, 595, 637, 645, 651, 671, 679, 703, 715, 721, 763, 775, 781, 793, 805, 817, 841, 847, 861, 871, 889, 891, 925, 931, 949, 961, 973, 1001, 1015, 1027, 1035, 1045

Offset: 1

Author

Karsten Meyer, Dec 26 2010

Keywords

• nonn

Comments

Odd composite numbers q such that gcd(A000010(q), A000265(q-1)) > 1. - Robert Israel, Dec 20 2017

Examples

			18^3 == 1 mod 49
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000265.

Programs

Maple

filter:= proc(n)
  if isprime(n) then return false fi;
  igcd((n-1)/2^padic:-ordp(n-1,2), numtheory:-phi(n)) > 1
end proc:
select(filter, [seq(i,i=9..2000,2)]); # Robert Israel, Dec 20 2017

Mathematica

filterQ[n_] := If[PrimeQ[n], False, GCD[(n-1)/2^IntegerExponent[n-1, 2], EulerPhi[n]] > 1];
Select[Range[9, 2000, 2], filterQ] (* Jean-François Alcover, Sep 25 2020, after Robert Israel *)

Formula

a^d == 1 mod q

Extensions

Corrected by Robert Israel, Dec 20 2017