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A295584 Odd numbers that are not a product of Mersenne numbers (A000225).

%I A295584 #12 Sep 18 2018 09:09:52
%S A295584 5,11,13,17,19,23,25,29,33,35,37,39,41,43,47,51,53,55,57,59,61,65,67,
%T A295584 69,71,73,75,77,79,83,85,87,89,91,95,97,99,101,103,107,109,111,113,
%U A295584 115,117,119,121,123,125,129,131,133,137,139,141,143,145,149,151,153
%N A295584 Odd numbers that are not a product of Mersenne numbers (A000225).
%C A295584 Numbers m such that no commutative ring has m units.
%H A295584 Robert Israel, <a href="/A295584/b295584.txt">Table of n, a(n) for n = 1..10000</a>
%H A295584 S. Chebolu and K. Lockridge, <a href="http://dx.doi.org/10.4169/amer.math.monthly.124.10.960">How Many Units Can a Commutative Ring Have?</a>, Amer. Math. Monthly, 124 (2017), 960-965; <a href="https://arxiv.org/abs/1701.02341">arXiv</a>, arXiv:1701.02341 [math.AC], 2017.
%p A295584 N:= 1000: # to get all terms <= N
%p A295584 P:= {1}:
%p A295584 for k from 2 do
%p A295584   m:= 2^k-1;
%p A295584   if m > N then break fi;
%p A295584   P:= map(p -> seq(p*m^j, j=0..floor(log[m](N/p))), P);
%p A295584 od:
%p A295584 sort(convert({seq(i,i=1..N,2)} minus P, list)); # _Robert Israel_, Dec 15 2017
%t A295584 nn == 1000;
%t A295584 P = {1};
%t A295584 For[k = 2, True, k++,
%t A295584    m = 2^k - 1;
%t A295584    If[m > nn, Break[]
%t A295584 ];
%t A295584 P = (Function[p, Table[p m^j, {j, 0, Log[m, nn/p]}]] /@ P) // Flatten];
%t A295584 Range[1, nn, 2] ~Complement~ P (* _Jean-François Alcover_, Sep 18 2018, after _Robert Israel_ *)
%Y A295584 Odd numbers not in A282572; complement of A296241.
%Y A295584 Cf. A000225.
%K A295584 nonn
%O A295584 1,1
%A A295584 _N. J. A. Sloane_, Dec 15 2017