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A064027 a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.

FORMULA

Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.

a(n) = (-1)^n*(A001157(n) - 2*A050999(n)).

Logarithmic derivative of A224364. - Paul D. Hanna, Apr 04 2013

Bisection: a(2*k-1) = A001157(2*k-1), a(2*k) = 4*A001157(k) - A050999(2*k), k >= 1. In the Hardy reference a(n) = sigma^*_2(n). - Wolfdieter Lang, Jan 07 2017

G.f.: Sum_{k>=1} k^2*x^k/(1 - (-x)^k). - Ilya Gutkovskiy, Nov 09 2018

Conjecture: Sum_{k=1..n} a(k) ~ 7 * Zeta(3) * n^3 / 24. - Vaclav Kotesovec, Nov 10 2018

EXAMPLE

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

MATHEMATICA

a[n_] := (-1)^n DivisorSum[n, (-1)^# * #^2 &]; Array[a, 50] (* Jean-Fran├žois Alcover, Dec 23 2015 *)

PROG

(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

CROSSREFS

Cf. A001157, A002129, A008457, A050999, A224364.

Cf. A321543 - A321565, A321807 - A321836 for related sequences.

Sequence in context: A074178 A178996 A127852 * A321543 A212456 A028878

Adjacent sequences:  A064024 A064025 A064026 * A064028 A064029 A064030

KEYWORD

mult,nonn

AUTHOR

Vladeta Jovovic, Sep 11 2001

STATUS

approved

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Last modified May 21 15:32 EDT 2019. Contains 323444 sequences. (Running on oeis4.)