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A352050 Sum of the 4th powers of the divisor complements of the odd proper divisors of n.

%I A352050 #24 Oct 13 2023 06:52:31
%S A352050 0,16,81,256,625,1312,2401,4096,6642,10016,14641,20992,28561,38432,
%T A352050 51331,65536,83521,106288,130321,160256,196963,234272,279841,335872,
%U A352050 391250,456992,538083,614912,707281,821312,923521,1048576,1200643,1336352,1503651,1700608,1874161
%N A352050 Sum of the 4th powers of the divisor complements of the odd proper divisors of n.
%H A352050 Robert Israel, <a href="/A352050/b352050.txt">Table of n, a(n) for n = 1..10000</a>
%F A352050 a(n) = n^4 * Sum_{d|n, d<n, d odd} 1 / d^4.
%F A352050 G.f.: Sum_{k>=2} k^4 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 14 2023
%F A352050 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352050 a(n) = A051001(n) * A006519(n)^4 - A000035(n).
%F A352050 Sum_{k=1..n} a(k) = c * n^5 / 5, where c = 31*zeta(5)/32 = 1.00452376... . (End)
%e A352050 a(10) = 10^4 * Sum_{d|10, d<10, d odd} 1 / d^4 = 10^4 * (1/1^4 + 1/5^4) = 10016.
%p A352050 f:= proc(n) local m,d;
%p A352050       m:= n/2^padic:-ordp(n,2);
%p A352050       add((n/d)^4, d = select(`<`,numtheory:-divisors(m),n))
%p A352050 end proc:map(f, [$1..40]); # _Robert Israel_, Apr 03 2023
%t A352050 A352050[n_]:=DivisorSum[n,1/#^4&,#<n&&OddQ[#]&]n^4;Array[A352050,50] (* _Paolo Xausa_, Aug 09 2023 *)
%t A352050 a[n_] := DivisorSigma[-4, n/2^IntegerExponent[n, 2]] * n^4 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352050 (PARI) a(n) = n^4 * sigma(n >> valuation(n, 2), -4) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352050 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), A352049 (k=3), this sequence (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
%Y A352050 Cf. A000035, A006519, A013663, A051001.
%K A352050 nonn,easy
%O A352050 1,2
%A A352050 _Wesley Ivan Hurt_, Mar 01 2022