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

Author

Rémy Sigrist, Feb 16 2021

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, C program for A341285

Crossrefs

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

Programs

C

See Links section.

Formula

a(n) <= n.

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

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

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