A341309 - OEIS

%I #26 Mar 25 2023 12:08:04

%S 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,

%T 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,

%U 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

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

%C 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

%C Conjecture 2: row sums of A352269. - _Omar E. Pol_, Mar 18 2022

%H Paolo Xausa, <a href="/A341309/b341309.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A204217(n) - A352446(n), conjectured. - _Omar E. Pol_, Mar 16 2022

%t A341309[n_]:=With[{t=Floor[(Sqrt[8n+1]-1)/2]},DivisorSum[n,#&,OddQ[#]&&#<=t&]];

%t Array[A341309,100] (* _Paolo Xausa_, Mar 25 2023 *)

%o (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

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

%K nonn

%O 1,6

%A _N. J. A. Sloane_, Feb 14 2021