A366066 a(n) is the largest positive integer k such that n can be expressed as the sum of k distinct positive integers that are coprime to each other.

0, 1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 9, 9, 9

Offset

0,4

Comments

The indices at which k first appears, for k >= 0: 1, 3, 6, 11, 18, 29, 42, 59, 78 (A014284). Such n's are expressed as the sum of 1 and the first primes.

Runs with length >= 2 start at numbers k^2 - 1 (k >= 2).

If there are terms between runs of k and k+1, these two numbers occur alternately. Suppose that m is such a term that is b(m) terms after the first occurrence of k+1; if b(m) is odd, there are at least two even numbers in the expression of n as the sum of k+1 integers, which are not coprime to each other, so a(m) = k.

Links

Yifan Xie, Table of n, a(n) for n = 0..9999

Formula

a(n) = A083375(n) - 1 if and only if n = 7, 12, 14, 19, 21, 23, 30, 32, 34, 43, 45, 47, 60, 62, 79; otherwise, a(n) = A083375(n).

Example

For n = 11, 1+2+3+5=11; so a(11) = 4.
For n = 12, 1+4+7=12; so a(12) = 3.

Prog

(PARI) lista(nn) = v=[0]; f=[7, 12, 14, 19, 21, 23, 30, 32, 34, 43, 45, 47, 60, 62, 79]; for(n=1, nn, for(i=1, prime(n), v=concat(v, n))); for(n=1, 15, v[f[n]+1]=v[f[n]+1]-1); v;

Crossrefs

Cf. A000040, A000290, A014284, A083375.

Sequence in context: A350240 A342739 A137734 * A352627 A182210 A078705

Adjacent sequences: A366063 A366064 A366065 * A366067 A366068 A366069

Keyword

nonn,easy

Author

Yifan Xie, Sep 28 2023

Status

approved