A341309 Sum of odd divisors of n that are <= A003056(n).

1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 1, 4, 8, 1, 9, 1, 1, 4, 1, 13, 4, 1, 1, 4, 6, 1, 11, 1, 1, 18, 1, 1, 4, 8, 6, 4, 1, 1, 13, 6, 8, 4, 1, 1, 9, 1, 1, 20, 1, 6, 15, 1, 1, 4, 13, 1, 13, 1, 1, 9, 1, 19, 4, 1, 6, 13, 1, 1, 11, 6, 1, 4, 12, 1, 18, 21

Offset: 1

Author

N. J. A. Sloane, Feb 14 2021

Keywords

• nonn

Comments

Conjecture 1: a(n) is also the total number de parts in all partitions of n into an odd number of consecutive parts. - Omar E. Pol, Mar 16 2022

Conjecture 2: row sums of A352269. - Omar E. Pol, Mar 18 2022

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000593, A003056, A082647, A131576, A333807, A341310.

Programs

Mathematica

A341309[n_]:=With[{t=Floor[(Sqrt[8n+1]-1)/2]},DivisorSum[n,#&,OddQ[#]&&#<=t&]];
Array[A341309,100] (* Paolo Xausa, Mar 25 2023 *)

PARI

a(n) = my(m=n>>valuation(n, 2), s=(sqrtint(8*n+1)-1)\2); sumdiv(m, d, if (d <= s, d)); \\ Michel Marcus, Mar 25 2023

Formula

a(n) = A204217(n) - A352446(n), conjectured. - Omar E. Pol, Mar 16 2022