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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.
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%I #51 Sep 27 2017 09:59:05

%S 230,341,1387,2047,2701,3277,4033,4369,4681,5461,7957,8321,10261,

%T 13747,14491,15709,18721,19951,23377,31323,31417,31609,31621,35333,

%U 38193,42799,49141,49981,60701,60787,65077,65281,80581,83333,85489

%N 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.

%C 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.

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

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

%H Max Alekseyev, <a href="/A291617/b291617.txt">Table of n, a(n) for n = 1..66</a>

%H A. Rotkiewicz, <a href="https://eudml.org/doc/140812">Sur les nombres pseudopremiers de la forme M_p M_q</a>, Elemente der Mathematik 20 (1965): 108-109. (in French)

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

%t 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 *)

%o (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; }

%Y Cf. A001567, A050217, A214305.

%K nonn

%O 1,1

%A _Max Alekseyev_ and _Thomas Ordowski_, Aug 28 2017