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A328667
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a(n) = Sum_{d divides n} (-1)^(n + 1 + d + n/d) * d^2.
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2
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1, 5, 10, 13, 26, 50, 50, 45, 91, 130, 122, 130, 170, 250, 260, 173, 290, 455, 362, 338, 500, 610, 530, 450, 651, 850, 820, 650, 842, 1300, 962, 685, 1220, 1450, 1300, 1183, 1370, 1810, 1700, 1170, 1682, 2500, 1850, 1586, 2366, 2650, 2210, 1730, 2451, 3255
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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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Multiplicative with a(2^e) = (2^(2*e+1) + 7)/3 = A321358(e) if e>0, else a(p^e) = (p^(2*e+2) - 1)/(p^2 - 1).
G.f.: Sum_{k>=1} k^2 * x^k/(1 + (-x)^k) = Sum_{k>=1} x^k*(1 - (-x)^k)/(1 + (-x)^k)^3.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 7*zeta(3)/24 = 0.350599... . - Amiram Eldar, Nov 01 2022
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EXAMPLE
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G.f. = x + 5*x^2 + 10*x^3 + 13*x^4 + 26*x^5 + 50*x^6 + 50*x^7 + 45*x^8 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, DivisorSum[n, (-1)^(n + 1 + # + n/#) #^2 &];
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PROG
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(PARI) {a(n) = sumdiv(n, d, (-1)^(n + 1 + n\d + d)*d^2)};
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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