A347737 Zero together with the partial sums of A238005.

0, 0, 0, 0, 1, 1, 2, 3, 5, 5, 7, 9, 11, 13, 15, 16, 20, 23, 25, 28, 31, 33, 37, 41, 45, 48, 52, 54, 59, 64, 67, 72, 78, 81, 86, 89, 94, 100, 106, 110, 116, 122, 126, 132, 138, 141, 148, 155, 162, 168, 174, 179, 186, 193, 198, 204, 212, 218, 226, 234, 240, 248, 256, 260

Offset

0,7

Comments

a(n) is also the total number of ones in the first n rows of A347579, n >= 1.

Links

Table of n, a(n) for n=0..63.

Formula

a(n) = A006463(n+1) - A060831(n).

Mathematica

Accumulate@Table[Length@Select[Select[IntegerPartitions@n, DuplicateFreeQ], Differences@MinMax@#=={Length@#}&], {n, 60}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

Prog

(Python)
from math import isqrt
def A347737(n): return (r:=isqrt((n+1<<3)+1)-1>>1)*(6*n+4-r*(r+3))//6-((t:=isqrt(m:=n>>1))+(s:=isqrt(n)))*(t-s)-(sum(n//k for k in range(1, s+1))-sum(m//k for k in range(1, t+1))<<1) # Chai Wah Wu, Jun 07 2025

Crossrefs

a(n) is also the total number of zeros in the first n rows of the triangles A196020, A211343, A231345, A236106, A237048 (simpler), A239662, A261699, A271344, A272026, A280850, A285574, A285891, A285914, A286013, A296508 (and possibly others), n >= 1.

Cf. A001227, A003056, A006463, A060831, A237593, A238005, A347579.

Sequence in context: A187322 A335429 A156899 * A332286 A357250 A156898

Adjacent sequences: A347734 A347735 A347736 * A347738 A347739 A347740

Keyword

nonn

Author

Omar E. Pol, Sep 11 2021

Status

approved