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A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.

%I A352052 #21 Oct 13 2023 06:52:19
%S A352052 0,64,729,4096,15625,46720,117649,262144,532170,1000064,1771561,
%T A352052 2990080,4826809,7529600,11406979,16777216,24137569,34058944,47045881,
%U A352052 64004096,85884499,113379968,148035889,191365120,244156250,308915840,387952659,481894400,594823321
%N A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.
%H A352052 Robert Israel, <a href="/A352052/b352052.txt">Table of n, a(n) for n = 1..10000</a>
%F A352052 a(n) = n^6 * Sum_{d|n, d<n, d odd} 1 / d^6.
%F A352052 G.f.: Sum_{k>=2} k^6 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 18 2023
%F A352052 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352052 a(n) = A321810(n) * A006519(n)^6 - A000035(n).
%F A352052 Sum_{k=1..n} a(k) = c * n^7 / 7, where c = 127*zeta(7)/128 = 1.000471548... . (End)
%e A352052 a(10) = 10^6 * Sum_{d|10, d<10, d odd} 1 / d^6 = 10^6 * (1/1^6 + 1/5^6) = 1000064.
%p A352052 f:= proc(n) local m,d;
%p A352052       m:= n/2^padic:-ordp(n,2);
%p A352052       add((n/d)^6, d = select(`<`,numtheory:-divisors(m),n))
%p A352052 end proc:
%p A352052 map(f, [$1..30]); # _Robert Israel_, Apr 03 2023
%t A352052 Table[n^6*DivisorSum[n, 1/#^6 &, And[# < n, OddQ[#]] &], {n, 29}] (* _Michael De Vlieger_, Apr 04 2023 *)
%t A352052 a[n_] := DivisorSigma[-6, n/2^IntegerExponent[n, 2]] * n^6 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352052 (PARI) a(n) = n^6*sumdiv(n, d, if ((d<n) && (d%2), 1/d^6)); \\ _Michel Marcus_, Apr 04 2023
%o A352052 (PARI) a(n) = n^6 * sigma(n >> valuation(n, 2), -6) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352052 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), A352051 (k=5), this sequence (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
%Y A352052 Cf. A000035, A006519, A013665, A321810.
%K A352052 nonn,easy
%O A352052 1,2
%A A352052 _Wesley Ivan Hurt_, Mar 01 2022