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

%I A291601 #22 Apr 22 2024 04:05:53
%S A291601 341,1105,1387,2047,2701,3277,4033,4369,4681,5461,7957,8321,10261,
%T A291601 13747,13981,14491,15709,18721,19951,23377,31417,31609,31621,35333,
%U A291601 42799,49141,49981,60701,60787,65077,65281,68101,80581,83333,85489,88357,90751,104653,123251,129889
%N A291601 Composite integers k such that 2^d == 2^(k/d) (mod k) for all d|k.
%C A291601 Such k must be odd.
%C A291601 For d=1, we have 2^k == 2 (mod k), implying that k is a Fermat pseudoprime (A001567).
%C A291601 Every Super-Poulet number belongs to this sequence.
%H A291601 Amiram Eldar, <a href="/A291601/b291601.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Robert Israel)
%p A291601 filter:= proc(n) local D,d;
%p A291601   if isprime(n) then return false fi;
%p A291601   D:= sort(convert(numtheory:-divisors(n),list));
%p A291601   for d in D while d^2 < n do
%p A291601     if 2 &^ d - 2 &^(n/d) mod n <> 0 then return false fi
%p A291601   od:
%p A291601   true
%p A291601 end proc:
%p A291601 select(filter, [seq(i,i=3..2*10^5,2)]); # _Robert Israel_, Aug 28 2017
%t A291601 filterQ[n_] := CompositeQ[n] && AllTrue[Divisors[n], PowerMod[2, #, n] == PowerMod[2, n/#, n]&];
%t A291601 Select[Range[1, 10^6, 2], filterQ] (* _Jean-François Alcover_, Jun 18 2020 *)
%o A291601 (PARI) is(k) = {if(k == 1 || !(k%2) || isprime(k), return(0)); fordiv(k, d, if(d^2 <= k && Mod(2, k)^d != Mod(2, k)^(k/d), return(0))); 1;} \\ _Amiram Eldar_, Apr 22 2024
%Y A291601 Subsequence of A001567.
%Y A291601 Supersequence of A050217, their set difference is given by A291602.
%Y A291601 Cf. A291602.
%K A291601 nonn
%O A291601 1,1
%A A291601 _Max Alekseyev_, Aug 27 2017