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A064027
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a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2.
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10
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1, 3, 10, 19, 26, 30, 50, 83, 91, 78, 122, 190, 170, 150, 260, 339, 290, 273, 362, 494, 500, 366, 530, 830, 651, 510, 820, 950, 842, 780, 962, 1363, 1220, 870, 1300, 1729, 1370, 1086, 1700, 2158, 1682, 1500, 1850, 2318, 2366, 1590, 2210, 3390, 2451, 1953
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OFFSET
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1,2
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REFERENCES
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G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
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LINKS
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FORMULA
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Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 1/2^(s-1) + 1/2^(2*s-3)). - Amiram Eldar, Sep 21 2023
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EXAMPLE
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L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + ...
where exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 + 106*x^8 + 181*x^9 + ... + A224364(n)*x^n + ... - Paul D. Hanna, Apr 04 2013
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MATHEMATICA
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a[n_] := If[OddQ[n], 1, (1 - 6/(4^(IntegerExponent[n, 2] + 1) - 1))] * DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)
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PROG
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(PARI) {a(n)=if(n<1, 0, (-1)^n*sumdiv(n^1, d, (-1)^d*d^2))} \\ Paul D. Hanna, Apr 04 2013
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k^2*x^k/(1-(-x)^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018
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CROSSREFS
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KEYWORD
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mult,easy,nonn
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AUTHOR
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STATUS
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approved
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