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A360779 Refactorable numbers gaps: differences between consecutive refactorable numbers.

%I A360779 #18 Mar 05 2023 17:09:05
%S A360779 1,6,1,3,6,6,12,4,16,4,12,8,4,4,8,8,4,20,4,4,16,4,24,4,20,21,3,4,8,8,
%T A360779 4,24,12,8,32,16,4,12,12,4,8,12,28,17,3,4,2,18,4,8,8,4,12,12,20,24,4,
%U A360779 4,16,16,12,13,7,4,4,24,8,12,24,4,8,12,44,16,12,4,16,4,24
%N A360779 Refactorable numbers gaps: differences between consecutive refactorable numbers.
%C A360779 Empirically it looks as though the consecutive refactorable numbers >= 8 with odd gaps between them always occur in triples: [8, 9, 12], [204, 225, 228], [424, 441, 444], [612, 625, 632], [1068, 1089, 1096], [1520, 1521, 1524], and so on. The sum of the gaps in the triple is divisible by 4. The middle term of a triple is an odd refactorable number, see A036896.
%F A360779 a(n) = A033950(n + 1) - A033950(n).
%e A360779 a(1) = 2 - 1 = 1;
%e A360779 a(2) = 8 - 2 = 6;
%e A360779 a(3) = 9 - 8 = 1;
%e A360779 and so on.
%t A360779 Differences[Select[Range[1000], Divisible[#, DivisorSigma[0, #]] &]] (* _Amiram Eldar_, Feb 20 2023 *)
%o A360779 (PARI) lista(nn) = my(v=select(x->!(x % numdiv(x)), [1..nn])); vector(#v-1, k, v[k+1]-v[k]); \\ _Michel Marcus_, Feb 20 2023
%Y A360779 Cf. A033950, A036896, A036898, A114617.
%K A360779 nonn
%O A360779 1,2
%A A360779 _Ctibor O. Zizka_, Feb 20 2023