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A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

%I A050217 #74 Feb 16 2025 08:32:40
%S A050217 341,1387,2047,2701,3277,4033,4369,4681,5461,7957,8321,10261,13747,
%T A050217 14491,15709,18721,19951,23377,31417,31609,31621,35333,42799,49141,
%U A050217 49981,60701,60787,65077,65281,80581,83333,85489,88357,90751
%N A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.
%C A050217 Every semiprime in A001567 is in this sequence (see Sierpiński). a(61) = 294409 is the first term having more than two prime factors. See A178997 for super-Poulet numbers having more than two prime factors. - _T. D. Noe_, Jan 11 2011
%C A050217 Composite numbers n such that 2^d == 2 (mod n) for every d|n. - _Thomas Ordowski_, Sep 04 2016
%C A050217 Composite numbers n such that 2^p == 2 (mod n) for every prime p|n. - _Thomas Ordowski_, Sep 06 2016
%C A050217 Composite numbers n = p(1)^e(1)*p(2)^e(2)*...*p(k)^e(k) such that 2^gcd(p(1)-1,p(2)-1,...,p(k)-1) == 1 (mod n). - _Thomas Ordowski_, Sep 12 2016
%C A050217 Nonsquarefree terms are divisible by the square of a Wieferich prime (see A001220). These include 1194649, 12327121, 5654273717, 26092328809, 129816911251. - _Robert Israel_, Sep 13 2016
%C A050217 Composite numbers n such that 2^A258409(n) == 1 (mod n). - _Thomas Ordowski_, Sep 15 2016
%D A050217 W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964, p. 231.
%H A050217 Amiram Eldar, <a href="/A050217/b050217.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A050217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Super-PouletNumber.html">Super-Poulet Numbers</a>
%H A050217 Wikipedia, <a href="http://en.wikipedia.org/wiki/Super-Poulet_number">Super-Poulet number</a>
%p A050217 filter:= = proc(n)
%p A050217     not isprime(n) and andmap(p -> 2&^p mod n = 2, numtheory:-factorset(n))
%p A050217 end proc:
%p A050217 select(filter, [seq(i,i=3..10^5,2)]); # _Robert Israel_, Sep 13 2016
%t A050217 Select[Range[1, 110000, 2], !PrimeQ[#] && Union[PowerMod[2, Rest[Divisors[#]], #]] == {2} & ]
%o A050217 (PARI) is(n)=if(isprime(n), return(0)); fordiv(n,d, if(Mod(2,d)^d!=2, return(0))); n>1 \\ _Charles R Greathouse IV_, Aug 27 2016
%Y A050217 A214305 is a subsequence.
%Y A050217 A065341 is a subsequence. - _Thomas Ordowski_, Nov 20 2016
%Y A050217 Cf. A001220, A001567, A178997.
%K A050217 nonn
%O A050217 1,1
%A A050217 _Eric W. Weisstein_