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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

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OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..79.

FORMULA

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

EXAMPLE

a(1) = 2 - 1 = 1;

a(2) = 8 - 2 = 6;

a(3) = 9 - 8 = 1;

and so on.

MATHEMATICA

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

PROG