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A352051 Sum of the 5th powers of the divisor complements of the odd proper divisors of n.

%I A352051 #20 Oct 13 2023 06:52:13
%S A352051 0,32,243,1024,3125,7808,16807,32768,59292,100032,161051,249856,
%T A352051 371293,537856,762743,1048576,1419857,1897376,2476099,3201024,4101151,
%U A352051 5153664,6436343,7995392,9768750,11881408,14408199,17211392,20511149,24407808,28629151,33554432,39296687
%N A352051 Sum of the 5th powers of the divisor complements of the odd proper divisors of n.
%H A352051 Robert Israel, <a href="/A352051/b352051.txt">Table of n, a(n) for n = 1..10000</a>
%F A352051 a(n) = n^5 * Sum_{d|n, d<n, d odd} 1 / d^5.
%F A352051 G.f.: Sum_{k>=2} k^5 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 18 2023
%F A352051 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352051 a(n) = A051002(n) * A006519(n)^5 - A000035(n).
%F A352051 Sum_{k=1..n} a(k) = c * n^6 / 6, where c = 63*zeta(6)/64 = 1.00144707... . (End)
%e A352051 a(10) = 10^5 * Sum_{d|10, d<10, d odd} 1 / d^5 = 10^5 * (1/1^5 + 1/5^5) = 100032.
%p A352051 f:= proc(n) local m,d;
%p A352051       m:= n/2^padic:-ordp(n,2);
%p A352051       add((n/d)^5, d = select(`<`,numtheory:-divisors(m),n))
%p A352051 end proc:
%p A352051 map(f, [$1..40]); # _Robert Israel_, Apr 03 2023
%t A352051 A352051[n_]:=DivisorSum[n,1/#^5&,#<n&&OddQ[#]&]n^5;Array[A352051,50] (* _Paolo Xausa_, Aug 09 2023 *)
%t A352051 a[n_] := DivisorSigma[-5, n/2^IntegerExponent[n, 2]] * n^5 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352051 (PARI) a(n) = n^5 * sigma(n >> valuation(n, 2), -5) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352051 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), A352050 (k=4), this sequence (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
%Y A352051 Cf. A000035, A006519, A013664, A051002.
%K A352051 nonn,easy
%O A352051 1,2
%A A352051 _Wesley Ivan Hurt_, Mar 01 2022