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A321823
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a(n) = Sum_{d|n, d==1 mod 4} d^7 - Sum_{d|n, d==3 mod 4} d^7.
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3
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1, 1, -2186, 1, 78126, -2186, -823542, 1, 4780783, 78126, -19487170, -2186, 62748518, -823542, -170783436, 1, 410338674, 4780783, -893871738, 78126, 1800262812, -19487170, -3404825446, -2186, 6103593751, 62748518, -10455572420, -823542
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^7*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^7)^(e+1)-1)/(p^7-1) if p == 1 (mod 4) and ((-p^7)^(e+1)-1)/(-p^7-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
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MATHEMATICA
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s[n_, r_] := DivisorSum[n, #^7 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^7)^(e+1)-1)/(p^7-1), ((-p^7)^(e+1)-1)/(-p^7-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
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PROG
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(PARI) apply( A321823(n)=sumdiv(n>>valuation(n, 2), d, (2-d%4)*d^7), [1..40]) \\ M. F. Hasler, Nov 26 2018
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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