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A308065 Nonnegative integers that are not the sum of two refactorable numbers whose difference is refactorable.

%I A308065 #37 Jul 31 2025 08:41:05
%S A308065 0,1,2,4,5,6,7,8,9,11,12,13,14,15,16,18,19,20,21,22,23,24,25,26,28,29,
%T A308065 30,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,55,
%U A308065 56,57,58,59,61,62,63,64,65,66,67,68,69,70,71,72,73,74
%N A308065 Nonnegative integers that are not the sum of two refactorable numbers whose difference is refactorable.
%C A308065 Differs from A047318 (not congruent to 3 (mod 7)) starting at a(n=22) = 24 = 3*7 + 3 which is in this sequence but not in A047318. - _M. F. Hasler_, Jun 30 2025
%H A308065 Robert Israel, <a href="/A308065/b308065.txt">Table of n, a(n) for n = 1..10000</a>
%H A308065 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RefactorableNumber.html">Refactorable Number</a>
%p A308065 notref:= proc(n) option remember; n mod numtheory:-tau(n) <> 0 end proc:
%p A308065 filter:= proc(n)
%p A308065   andmap(t -> notref(t) or notref(n-t) or notref(n-2*t), [$1 .. (n-1)/2])
%p A308065 end proc:
%p A308065 select(filter, [$0..100]); # _Robert Israel_, Jul 29 2025
%t A308065 Flatten[Table[If[Sum[(1 - Ceiling[(n - 2 i)/DivisorSigma[0, n - 2 i]] + Floor[(n - 2 i)/DivisorSigma[0, n - 2 i]]) (1 - Ceiling[i/DivisorSigma[0, i]] + Floor[i/DivisorSigma[0, i]]) (1 - Ceiling[(n - i)/DivisorSigma[0, n - i]] + Floor[(n - i)/DivisorSigma[0, n - i]]), {i, Floor[(n - 1)/2]}] == 0, n, {}], {n, 0, 100}]]
%Y A308065 Cf. A033950, A309347.
%Y A308065 Different from A047318 (not congruent to 3 modulo 7) and A057904.
%K A308065 nonn
%O A308065 1,3
%A A308065 _Wesley Ivan Hurt_, Jul 27 2019