A291617 Numbers p_1*p_2*...*p_k such that (2^p_1-1)*(2^p_2-1)*...*(2^p_k-1) is a Poulet number (A001567), where p_i are primes and k >= 2.

230, 341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31323, 31417, 31609, 31621, 35333, 38193, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489

Offset: 1

Author

Max Alekseyev and Thomas Ordowski, Aug 28 2017

Keywords

• nonn

Comments

Rotkiewicz (1965) proved that (2^p-1)*(2^q-1) is a Poulet number if and only if p*q is a Poulet number, where p,q are distinct primes. It follows that this sequence contains all nonsquare terms in A214305.

Generally, the sequence includes all squarefree super-Poulet numbers.

The terms n = 230, 31323, 38193, ... are not in A050217. Are there infinitely many such terms?

Examples

			The number n = 341 = 11*31 is a term, because m = (2^11-1)*(2^31-1) = 4395899025409 is a Poulet number.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..66
• A. Rotkiewicz, Sur les nombres pseudopremiers de la forme M_p M_q, Elemente der Mathematik 20 (1965): 108-109. (in French)

Crossrefs

Cf. A001567, A050217, A214305.

Programs

Mathematica

Select[Select[Range[10^4], CompositeQ@ # && SquareFreeQ@ # &], ! PrimeQ[#] && PowerMod[2, (# - 1), #] == 1 &@ Apply[Times, Map[2^# - 1 &, FactorInteger[#][[All, 1]] ]] &] (* Michael De Vlieger, Aug 30 2017 *)

PARI

{ is_A291617(n) = my(p,m); if(isprime(n),return(0)); p=factor(n); m=prod(i=1,matsize(p)[1], (2^p[i,1]-1)^p[i,2]); Mod(2,m)^m==2; }