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

%I A341309 #26 Mar 25 2023 12:08:04
%S A341309 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 A341309 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 A341309 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 A341309 Sum of odd divisors of n that are <= A003056(n).
%C A341309 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 A341309 Conjecture 2: row sums of A352269. - _Omar E. Pol_, Mar 18 2022
%H A341309 Paolo Xausa, <a href="/A341309/b341309.txt">Table of n, a(n) for n = 1..10000</a>
%F A341309 a(n) = A204217(n) - A352446(n), conjectured. - _Omar E. Pol_, Mar 16 2022
%t A341309 A341309[n_]:=With[{t=Floor[(Sqrt[8n+1]-1)/2]},DivisorSum[n,#&,OddQ[#]&&#<=t&]];
%t A341309 Array[A341309,100] (* _Paolo Xausa_, Mar 25 2023 *)
%o A341309 (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 A341309 Cf. A000593, A003056, A082647, A131576, A333807, A341310.
%K A341309 nonn
%O A341309 1,6
%A A341309 _N. J. A. Sloane_, Feb 14 2021