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A365475 a(n) is the first odd prime p such that A000120((2^n-1)*p) = n * A000120(p).

%I A365475 #7 Sep 04 2023 12:26:45
%S A365475 3,5,17,17,257,257,257,257,65537,65537,65537,65537,65537,65537,65537,
%T A365475 65537,4398054899713,4398054899713,4398054899713,1125899915231233,
%U A365475 1125899915231233,1125899915231233,1125899915231233,2251799847239681,2251799847239681,1152921513196781569,1152921513196781569
%N A365475 a(n) is the first odd prime p such that A000120((2^n-1)*p) = n * A000120(p).
%C A365475 a(n) is the first odd prime p such that between any two 1's in the base-2 representation of p there are at least n-1 0's.
%H A365475 Robert Israel, <a href="/A365475/b365475.txt">Table of n, a(n) for n = 1..1000</a>
%e A365475 a(3) = 17 because A000120((2^3-1) * 17) = A000120(119) = 6 = 3 * A000120(17).
%p A365475 f:= proc(m) local S,d,j,r,q;
%p A365475   S[0]:= [1];
%p A365475   for d from 1 do
%p A365475     S[d]:= NULL;
%p A365475     for j from 0 to d-m do
%p A365475       for r in S[j] do
%p A365475         q:= r + 2^d;
%p A365475         if isprime(q) then return q fi;
%p A365475         S[d]:= S[d],q;
%p A365475       od;
%p A365475     od;
%p A365475     S[d]:= [S[d]];
%p A365475   od;
%p A365475 end proc:
%p A365475 map(f, [$1..30]);
%Y A365475 Cf. A000120.
%K A365475 nonn,base
%O A365475 1,1
%A A365475 _Robert Israel_, Sep 04 2023