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A291601 Composite integers n such that 2^d == 2^(n/d) (mod n) for all d|n. 3


%S 341,1105,1387,2047,2701,3277,4033,4369,4681,5461,7957,8321,10261,

%T 13747,13981,14491,15709,18721,19951,23377,31417,31609,31621,35333,

%U 42799,49141,49981,60701,60787,65077,65281,68101,80581,83333,85489,88357,90751,104653,123251,129889

%N Composite integers n such that 2^d == 2^(n/d) (mod n) for all d|n.

%C Such n must be odd.

%C For d=1, we have 2^n == 2 (mod n), implying that n is a Fermat pseudoprime (A001567).

%C Every Super-Poulet number belongs to this sequence.

%H Robert Israel, <a href="/A291601/b291601.txt">Table of n, a(n) for n = 1..1000</a>

%p filter:= proc(n) local D,d;

%p if isprime(n) then return false fi;

%p D:= sort(convert(numtheory:-divisors(n),list));

%p for d in D while d^2 < n do

%p if 2 &^ d - 2 &^(n/d) mod n <> 0 then return false fi

%p od:

%p true

%p end proc:

%p select(filter, [seq(i,i=3..2*10^5,2)]); # _Robert Israel_, Aug 28 2017

%t filterQ[n_] := CompositeQ[n] && AllTrue[Divisors[n], PowerMod[2, #, n] == PowerMod[2, n/#, n]&];

%t Select[Range[1, 10^6, 2], filterQ] (* _Jean-Fran├žois Alcover_, Jun 18 2020 *)

%Y Subsequence of A001567.

%Y Supersequence of A050217, their set difference is given by A291602.

%K nonn

%O 1,1

%A _Max Alekseyev_, Aug 27 2017

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Last modified July 12 04:12 EDT 2020. Contains 335658 sequences. (Running on oeis4.)