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

0,2

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

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.

Example

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.

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 *)

Crossrefs

Cf. A019444.

Sequence in context: A123664 A084980 A101369 * A103864 A268712 A152679

Adjacent sequences: A125144 A125145 A125146 * A125148 A125149 A125150

Keyword

nonn

Author

Leroy Quet, Jan 11 2007

Extensions

Extended by Ray Chandler, Jan 23 2007

Status

approved