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A344721
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a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^3.
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4
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1, 7, 27, 56, 118, 196, 324, 448, 685, 901, 1233, 1549, 2019, 2445, 3157, 3664, 4482, 5262, 6290, 7128, 8536, 9598, 11118, 12392, 14255, 15743, 18087, 19711, 22149, 24417, 27209, 29251, 32771, 35327, 39087, 42048, 46046, 49244, 54180, 57512, 62434, 66838, 72258, 76246
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^3 - (d - 1)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 + x^k).
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MATHEMATICA
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a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-3*DivisorSigma[0, n] + 2*DivisorSigma[0, 2*n] + 6*DivisorSigma[1, n] - 3*DivisorSigma[1, 2*n] - 9/2 * DivisorSigma[2, n] + 3/2 * DivisorSigma[2, 2*n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)
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PROG
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(PARI) a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^3);
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^3-(d-1)^3)));
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1+x^k))/(1-x))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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