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A352049 Sum of the cubes of the divisor complements of the odd proper divisors of n.

%I A352049 #23 Oct 13 2023 06:52:26
%S A352049 0,8,27,64,125,224,343,512,756,1008,1331,1792,2197,2752,3527,4096,
%T A352049 4913,6056,6859,8064,9631,10656,12167,14336,15750,17584,20439,22016,
%U A352049 24389,28224,29791,32768,37295,39312,43343,48448,50653,54880,61543,64512,68921,77056,79507
%N A352049 Sum of the cubes of the divisor complements of the odd proper divisors of n.
%H A352049 Robert Israel, <a href="/A352049/b352049.txt">Table of n, a(n) for n = 1..10000</a>
%F A352049 a(n) = n^3 * Sum_{d|n, d<n, d odd} 1 / d^3.
%F A352049 G.f.: Sum_{k>=2} k^3 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 14 2023
%F A352049 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352049 a(n) = A051000(n) * A006519(n)^3 - A000035(n).
%F A352049 Sum_{k=1..n} a(k) = c * n^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)
%e A352049 a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.
%p A352049 f:= proc(n) local m,d;
%p A352049       m:= n/2^padic:-ordp(n,2);
%p A352049       add((n/d)^3, d = select(`<`,numtheory:-divisors(m),n))
%p A352049 end proc:
%p A352049 map(f, [$1..50]); # _Robert Israel_, Apr 03 2023
%t A352049 A352049[n_]:=DivisorSum[n,1/#^3&,#<n&&OddQ[#]&]n^3;Array[A352049,50] (* _Paolo Xausa_, Aug 09 2023 *)
%t A352049 a[n_] := DivisorSigma[-3, n/2^IntegerExponent[n, 2]] * n^3 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352049 (PARI) a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352049 Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), this sequence (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
%Y A352049 Cf. A006519, A051000, A300707.
%K A352049 nonn,easy
%O A352049 1,2
%A A352049 _Wesley Ivan Hurt_, Mar 01 2022