OFFSET
1,1
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?
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)
EXAMPLE
The number n = 341 = 11*31 is a term, because m = (2^11-1)*(2^31-1) = 4395899025409 is a Poulet number.
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 *)
PROG
(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; }
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Alekseyev and Thomas Ordowski, Aug 28 2017
STATUS
approved