A352049 - OEIS

%I #23 Oct 13 2023 06:52:26

%S 0,8,27,64,125,224,343,512,756,1008,1331,1792,2197,2752,3527,4096,

%T 4913,6056,6859,8064,9631,10656,12167,14336,15750,17584,20439,22016,

%U 24389,28224,29791,32768,37295,39312,43343,48448,50653,54880,61543,64512,68921,77056,79507

%N Sum of the cubes of the divisor complements of the odd proper divisors of n.

%H Robert Israel, <a href="/A352049/b352049.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n^3 * Sum_{d|n, d<n, d odd} 1 / d^3.

%F G.f.: Sum_{k>=2} k^3 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 14 2023

%F From _Amiram Eldar_, Oct 13 2023: (Start)

%F a(n) = A051000(n) * A006519(n)^3 - A000035(n).

%F Sum_{k=1..n} a(k) = c * n^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)

%e a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.

%p f:= proc(n) local m,d;

%p       m:= n/2^padic:-ordp(n,2);

%p       add((n/d)^3, d = select(`<`,numtheory:-divisors(m),n))

%p end proc:

%p map(f, [$1..50]); # _Robert Israel_, Apr 03 2023

%t A352049[n_]:=DivisorSum[n,1/#^3&,#<n&&OddQ[#]&]n^3;Array[A352049,50] (* _Paolo Xausa_, Aug 09 2023 *)

%t a[n_] := DivisorSigma[-3, n/2^IntegerExponent[n, 2]] * n^3 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)

%o (PARI) a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - n % 2; \\ _Amiram Eldar_, Oct 13 2023

%Y 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 Cf. A006519, A051000, A300707.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Mar 01 2022