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A352048 Sum of the squares of the divisor complements of the odd proper divisors of n. 11
0, 4, 9, 16, 25, 40, 49, 64, 90, 104, 121, 160, 169, 200, 259, 256, 289, 364, 361, 416, 499, 488, 529, 640, 650, 680, 819, 800, 841, 1040, 961, 1024, 1219, 1160, 1299, 1456, 1369, 1448, 1699, 1664, 1681, 2000, 1849, 1952, 2365, 2120, 2209, 2560, 2450, 2604, 2899, 2720 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
a(n) = n^2 * Sum_{d|n, d<n, d odd} 1 / d^2.
G.f.: Sum_{k>=2} k^2 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023
From Amiram Eldar, Oct 13 2023: (Start)
a(n) = A050999(n) * A006519(n)^2 - A000035(n).
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = 7*zeta(3))/8 = 1.0517997... (A233091). (End)
EXAMPLE
a(10) = 10^2 * Sum_{d|10, d<10, d odd} 1 / d^2 = 10^2 * (1/1^2 + 1/5^2) = 104.
MAPLE
f:= proc(n) local m, d;
m:= n/2^padic:-ordp(n, 2);
add((n/d)^2, d = select(`<`, numtheory:-divisors(m), n))
end proc:
map(f, [$1..60]); # Robert Israel, Apr 03 2023
MATHEMATICA
a[n_] := n^2 DivisorSum[n, If[# < n && OddQ[#], 1/#^2, 0]&];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)
a[n_] := DivisorSigma[-2, n/2^IntegerExponent[n, 2]] * n^2 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n^2*sumdiv(n, d, if ((d<n) && (d%2), 1/d^2)); \\ Michel Marcus, May 11 2023
(PARI) a(n) = n^2 * sigma(n >> valuation(n, 2), -2) - n % 2; \\ Amiram Eldar, Oct 13 2023
CROSSREFS
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).
Sequence in context: A019571 A008024 A008056 * A206920 A108612 A065741
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved

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Last modified March 4 16:26 EST 2024. Contains 370532 sequences. (Running on oeis4.)