cp's OEIS Frontend

A352048 Sum of the squares of the divisor complements of the odd proper divisors of n.

%I A352048 #33 Dec 25 2024 04:06:32
%S A352048 0,4,9,16,25,40,49,64,90,104,121,160,169,200,259,256,289,364,361,416,
%T A352048 499,488,529,640,650,680,819,800,841,1040,961,1024,1219,1160,1299,
%U A352048 1456,1369,1448,1699,1664,1681,2000,1849,1952,2365,2120,2209,2560,2450,2604,2899,2720
%N A352048 Sum of the squares of the divisor complements of the odd proper divisors of n.
%H A352048 Robert Israel, <a href="/A352048/b352048.txt">Table of n, a(n) for n = 1..10000</a>
%F A352048 a(n) = n^2 * Sum_{d|n, d<n, d odd} 1 / d^2.
%F A352048 G.f.: Sum_{k>=2} k^2 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 14 2023
%F A352048 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352048 a(n) = A050999(n) * A006519(n)^2 - A000035(n).
%F A352048 Sum_{k=1..n} a(k) = c * n^3 / 3, where c = 7*zeta(3)/8 = 1.0517997... (A233091). (End)
%e A352048 a(10) = 10^2 * Sum_{d|10, d<10, d odd} 1 / d^2 = 10^2 * (1/1^2 + 1/5^2) = 104.
%p A352048 f:= proc(n) local m,d;
%p A352048       m:= n/2^padic:-ordp(n,2);
%p A352048       add((n/d)^2, d = select(`<`,numtheory:-divisors(m),n))
%p A352048 end proc:
%p A352048 map(f, [$1..60]); # _Robert Israel_, Apr 03 2023
%t A352048 a[n_] := n^2 DivisorSum[n, If[# < n && OddQ[#], 1/#^2, 0]&];
%t A352048 Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, May 11 2023 *)
%t A352048 a[n_] := DivisorSigma[-2, n/2^IntegerExponent[n, 2]] * n^2 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352048 (PARI) a(n) = n^2*sumdiv(n, d, if ((d<n) && (d%2), 1/d^2)); \\ _Michel Marcus_, May 11 2023
%o A352048 (PARI) a(n) = n^2 * sigma(n >> valuation(n, 2), -2) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352048 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), this sequence (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
%Y A352048 Cf. A006519, A050999, A076577, A233091.
%K A352048 nonn,easy
%O A352048 1,2
%A A352048 _Wesley Ivan Hurt_, Mar 01 2022