A360779 Refactorable numbers gaps: differences between consecutive refactorable numbers.

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, 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, 4, 16, 16, 12, 13, 7, 4, 4, 24, 8, 12, 24, 4, 8, 12, 44, 16, 12, 4, 16, 4, 24

Offset: 1

Author

Ctibor O. Zizka, Feb 20 2023

Keywords

• nonn

Comments

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.

Examples

			a(1) = 2 - 1 = 1;
a(2) = 8 - 2 = 6;
a(3) = 9 - 8 = 1;
and so on.
		

Crossrefs

Cf. A033950, A036896, A036898, A114617.

Programs

Mathematica

Differences[Select[Range[1000], Divisible[#, DivisorSigma[0, #]] &]] (* Amiram Eldar, Feb 20 2023 *)

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

Formula

a(n) = A033950(n + 1) - A033950(n).