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A321831
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a(n) = Sum_{d|n, n/d==1 mod 4} d^7 - Sum_{d|n, n/d==3 mod 4} d^7.
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12
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1, 128, 2186, 16384, 78126, 279808, 823542, 2097152, 4780783, 10000128, 19487170, 35815424, 62748518, 105413376, 170783436, 268435456, 410338674, 611940224, 893871738, 1280016384, 1800262812, 2494357760, 3404825446, 4584374272, 6103593751
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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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Multiplicative with a(p^e) = round(p^(7e+7)/(p^7 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
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MATHEMATICA
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s[n_, r_] := DivisorSum[n, #^7 &, Mod[n/#, 4]==r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(7*e+7) - s[p]^(e+1))/(p^7 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
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PROG
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(PARI) apply( A321831(n)=factorback(apply(f->f[1]^(7*f[2]+7)\/(f[1]^7+f[1]%4-2), Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018
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CROSSREFS
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Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, this sequence, A321832, A321833, A321834, A321835, A321836.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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