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A284927
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a(n) = Sum_{d|n} (-1)^(n/d+1)*d^6.
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8
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1, 63, 730, 4031, 15626, 45990, 117650, 257983, 532171, 984438, 1771562, 2942630, 4826810, 7411950, 11406980, 16510911, 24137570, 33526773, 47045882, 62988406, 85884500, 111608406, 148035890, 188327590, 244156251, 304089030, 387952660, 474247150, 594823322
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OFFSET
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1,2
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COMMENTS
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Multiplicative because this sequence is the Dirichlet convolution of A001014 and A062157 which are both multiplicative. - Andrew Howroyd, Jul 20 2018
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LINKS
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FORMULA
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Multiplicative with a(2^e) = (31*2^(6*e+1)+1)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = 9*zeta(7)/64 = 0.141799... . (End)
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MATHEMATICA
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Table[Sum[(-1)^(n/d + 1)*d^6, {d, Divisors[n]}], {n, 50}] (* Indranil Ghosh, Apr 06 2017 *)
f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); f[2, e_] := (31*2^(6*e + 1) + 1)/63; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^6); \\ Indranil Ghosh, Apr 06 2017
(Python)
from sympy import divisors
print([sum([(-1)**(n//d + 1)*d**6 for d in divisors(n)]) for n in range(1, 51)]) # Indranil Ghosh, Apr 06 2017
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CROSSREFS
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Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), A284900 (k=4), A284926 (k=5), this sequence (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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