A291601 - OEIS

%I #22 Apr 22 2024 04:05:53

%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 k such that 2^d == 2^(k/d) (mod k) for all d|k.

%C Such k must be odd.

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

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

%H Amiram Eldar, <a href="/A291601/b291601.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Robert Israel)

%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-François Alcover_, Jun 18 2020 *)

%o (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 Subsequence of A001567.

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

%Y Cf. A291602.

%K nonn

%O 1,1

%A _Max Alekseyev_, Aug 27 2017