

A291617


Numbers p_1*p_2*...*p_k such that (2^p_11)*(2^p_21)*...*(2^p_k1) is a Poulet number (A001567), where p_i are primes and k >= 2.


1



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
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OFFSET

1,1


COMMENTS

Rotkiewicz (1965) proved that (2^p1)*(2^q1) 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 superPoulet numbers.
The terms n = 230, 31323, 38193, ... are not in A050217. Are there infinitely many such terms?


LINKS



EXAMPLE

The number n = 341 = 11*31 is a term, because m = (2^111)*(2^311) = 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



STATUS

approved



