A341285 Let B be the set of sequences of positive integers {b(k), k >= 0} such that for some k > 0 (necessarily unique) and any m >= 0, b(m+k) = b(m) + b(m+1) + ... + b(m+k-1); let g(b) = b(0); a(n) is the least value of g(b) for an element b of B containing n.

1, 2, 2, 3, 2, 4, 3, 2, 3, 4, 3, 4, 2, 4, 5, 4, 3, 3, 5, 6, 2, 6, 4, 4, 4, 4, 5, 5, 3, 5, 3, 5, 5, 2, 6, 6, 4, 4, 6, 6, 6, 4, 7, 4, 5, 7, 3, 7, 4, 5, 5, 6, 5, 8, 2, 6, 3, 6, 7, 4, 5, 6, 5, 5, 5, 6, 7, 4, 6, 4, 6, 6, 5, 8, 6, 3, 4, 6, 6, 6, 4, 8, 5, 7, 7, 6, 7

Offset

1,2

Comments

This sequence is a generalization of A249783 and A341456 to the set of "k-bonacci sequences of positive integers".

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, C program for A341285

Formula

a(n) <= n.

a(m*n) <= m*a(n).

a(n) = 2 iff n belongs to A020695.

a(n) = A070939(A341699(n)).

Example

The first terms of the elements b of B such that g(b) <= 3 are:
  g(b)  b(0)  b(1)  b(2)  b(3)  b(4)  b(5)  b(6)  b(7)  b(8)  b(9)
  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
     1     1     1     1     1     1     1     1     1     1     1
     2     2     2     2     2     2     2     2     2     2     2
     2     1     1     2     3     5     8    13    21    34    55
     3     3     3     3     3     3     3     3     3     3     3
     3     1     2     3     5     8    13    21    34    55    89
     3     2     1     3     4     7    11    18    29    47    76
     3     1     1     1     3     5     9    17    31    57   105
- so a(1) = 1,
     a(2) = a(3) = a(5) = a(8) = 2,
     a(4) = a(7) = a(9) = a(11) = a(17) = a(18) = 3.

Prog

(C) See Links section.

Crossrefs

Cf. A020695, A070939, A249783, A341456, A341699.

Sequence in context: A239514 A304464 A087050 * A263323 A263297 A163870

Adjacent sequences: A341282 A341283 A341284 * A341286 A341287 A341288

Keyword

nonn

Author

Rémy Sigrist, Feb 16 2021

Status

approved