A125147 a(0)=1; for n >= 1, a(n) is the smallest positive integer not occurring earlier in the sequence such that Sum_{k=0..n} a(k) is a multiple of n.

1, 2, 3, 6, 4, 9, 5, 12, 14, 7, 17, 8, 20, 22, 10, 25, 11, 28, 30, 13, 33, 35, 15, 38, 16, 41, 43, 18, 46, 19, 49, 51, 21, 54, 56, 23, 59, 24, 62, 64, 26, 67, 27, 70, 72, 29, 75, 77, 31, 80, 32, 83, 85, 34, 88, 90, 36, 93, 37, 96, 98, 39, 101, 40, 104, 106, 42, 109, 111, 44

Offset: 0

Author

Leroy Quet, Jan 11 2007

Keywords

• nonn

Comments

Sequence A019444 is a similar sequence, but it has an offset of 1 and a(1) =1 instead.

This sequence is a permutation of the positive integers. This can be proved using the theorem-prover Walnut. - Jeffrey Shallit, Aug 12 2023

Examples

			a(5) = 9 because 9 is the smallest positive integer m which does not occur earlier in the sequence and which is such that 5 divides m + Sum_{k=0..4} a(k).
So Sum_{k=0..5} a(k) = 25, which is divisible by 5.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 0..10000
• J. Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], August 12 2023.

Crossrefs

Cf. A019444.

Programs

Mathematica

f[l_List] := Block[{k = 1, n = Length[l], s = Plus @@ l},While[MemberQ[l, k] || Mod[s + k, n] > 0, k++ ];Append[l, k]];Nest[f, {1}, 70] (* Ray Chandler, Jan 23 2007 *)

Extensions

Extended by Ray Chandler, Jan 23 2007