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A279847 a(n) = Sum_{k=1..n} k^2*(floor(n/k) - 1). 1
0, 1, 2, 7, 8, 22, 23, 44, 54, 84, 85, 151, 152, 206, 241, 326, 327, 458, 459, 605, 664, 790, 791, 1065, 1091, 1265, 1356, 1622, 1623, 2023, 2024, 2365, 2496, 2790, 2865, 3480, 3481, 3847, 4026, 4636, 4637, 5373, 5374, 6000, 6341, 6875, 6876, 7982, 8032, 8787, 9086, 9952, 9953, 11137, 11284 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Sum of all squares of proper divisors of all positive integers <= n.

LINKS

Table of n, a(n) for n=1..55.

Index entries for sequences related to sums of divisors

FORMULA

G.f.: -x*(1 + x)/(1 - x)^4 + (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k).

a(n) = A064602(n) - A000330(n).

a(n) = Sum_{k=1..n} A067558(k).

a(n) = Sum_{k=1..n} (A001157(k) - A000290(k)).

a(p^k) = a(p^k-1) + (p^(2*k) - 1)/(p^2 - 1), for p is prime.

a(n) ~ ((zeta(3) - 1)/3)*n^3.

EXAMPLE

For n = 7 the proper divisors of the first seven positive integers are {0}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1} so a(7) = 0^2 + 1^2 + 1^2 + 1^2 + 2^2 + 1 ^2 + 1^2 + 2^2 + 3^2 + 1^2 = 23.

MATHEMATICA

Table[Sum[k^2 (Floor[n/k] - 1), {k, 1, n}], {n, 55}]

Table[Sum[DivisorSigma[2, k] - k^2, {k, 1, n}], {n, 55}]

PROG

(PARI) a(n) = sum(k=1, n, k^2*(floor(n/k)-1)) \\ Felix Fröhlich, Dec 20 2016

CROSSREFS

Cf. A000290, A000330, A001157, A064602, A067558, A153485.

Sequence in context: A026579 A167767 A054601 * A291629 A117558 A117559

Adjacent sequences:  A279844 A279845 A279846 * A279848 A279849 A279850

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Dec 20 2016

STATUS

approved

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Last modified June 20 09:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)