A357618 a(n) = sum of lengths of partitions of more than one consecutive positive integer adding up to n.

0, 0, 0, 2, 0, 2, 3, 2, 0, 5, 4, 2, 3, 2, 4, 10, 0, 2, 7, 2, 5, 11, 4, 2, 3, 7, 4, 11, 7, 2, 12, 2, 0, 11, 4, 14, 11, 2, 4, 11, 5, 2, 14, 2, 8, 25, 4, 2, 3, 9, 9, 11, 8, 2, 16, 17, 7, 11, 4, 2, 16, 2, 4, 27, 0, 17, 18, 2, 8, 11, 16

Offset

0,4

Comments

A polite number (A138591) has at least one partition of two or more consecutive positive integers that equals n. This sequence is the sum of lengths of all partitions that make a number polite.

This sequence is similar to A204217 which sums lengths of all partitions adding up to n including the partition of length 1.

Links

Table of n, a(n) for n=0..70.

Wikipedia, Polite number

Formula

a(n) = A204217(n) - 1 for n >= 1, a(0) = 0.

Example

n=15 is the sum of three partitions of n with two or more consecutive positive integers: 15 = 1 + 2 + 3 + 4 + 5, 15 = 4 + 5 + 6, 15 = 7 + 8.
The sum of the lengths of these partitions is a(15) = 5 + 3 + 2 = 10.
On the other hand a(8) = 0 because there are no partitions of two or more consecutive integers adding up to 8.

Prog

(Python)
def A357618(n):
  i=2; r=0
  while n//i>0:r+=(n%i==1)*i; n-=i; i+=1
  return r
A357618_list = [A357618(n) for n in range(70)]

Crossrefs

Cf. A069283 (politeness of a number), A138591 (polite numbers).

Cf. A204217.

Sequence in context: A264940 A272800 A104513 * A305610 A220455 A208295

Adjacent sequences: A357615 A357616 A357617 * A357619 A357620 A357621

Keyword

nonn,easy

Author

Daniel Vik, Oct 06 2022

Status

approved