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

341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609, 31621, 35333, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489, 88357, 90751

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

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

Composite numbers n such that 2^d == 2 (mod n) for every d|n. - Thomas Ordowski, Sep 04 2016

Composite numbers n such that 2^p == 2 (mod n) for every prime p|n. - Thomas Ordowski, Sep 06 2016

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

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

Composite numbers n such that 2^A258409(n) == 1 (mod n). - Thomas Ordowski, Sep 15 2016

References

• W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964, p. 231.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Eric Weisstein's World of Mathematics, Super-Poulet Numbers
• Wikipedia, Super-Poulet number

Crossrefs

A214305 is a subsequence.

A065341 is a subsequence. - Thomas Ordowski, Nov 20 2016

Cf. A001220, A001567, A178997.

Programs

Maple

filter:= = proc(n)
    not isprime(n) and andmap(p -> 2&^p mod n = 2, numtheory:-factorset(n))
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Sep 13 2016

Mathematica

Select[Range[1, 110000, 2], !PrimeQ[#] && Union[PowerMod[2, Rest[Divisors[#]], #]] == {2} & ]

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