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A321543
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a(n) = Sum_{d|n} (-1)^(d-1)*d^2.
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58
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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, 2900, -3230, 2810, -2460, 3172
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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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G.f.: Sum_{k>=1} (-1)^(k-1)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(1 - x^n)/(1 + x^n)^3. - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(2*e + 2) - 1)/3, and a(p^e) = (p^(2*e + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 04 2022
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MAPLE
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with(numtheory):
a := n -> add( (-1)^(d-1)*d^2, d in divisors(n) ): seq(a(n), n = 1..40);
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MATHEMATICA
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f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*e + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 04 2022 *)
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PROG
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(PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018
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CROSSREFS
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Glaisher's zeta_i, i=0..12: A048272, A002129, A321543, A138503, A279395, A321544, A321545, A321546, A321547, A321548, A321549, A321550, A321551.
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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