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A035202
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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 20.
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3
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1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 2, 1, 0, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 2, 1, 1, 2, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 0, 2, 0, 0, 1, 2, 1, 2, 0, 0, 0, 0
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OFFSET
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1,11
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COMMENTS
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Also number of divisors of n which end in 1 or 9 minus number of divisors of n which end in 3 or 7. E.g. a(98)=2-1=1 since divisors of 98 are: 1 and 49 counting +1 each; 2, 14 and 98 counting 0 each; and 7 counting -1. - Henry Bottomley, Jul 08 2003
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LINKS
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FORMULA
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a(n) = Sum_{d|n} Kronecker(20, d).
Multiplicative with a(p^e) = 1 if Kronecker(20, p) = 0 (p = 2 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(20, p) = -1 (p is in A003631 \ {2}), and a(p^e) = e+1 if Kronecker(20, p) = 1 (p is in A045468).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*log(phi)/sqrt(5) = 0.645613411446..., where phi is the golden ratio (A001622). (End)
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MAPLE
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a:= proc(n) local D, d; D:= map(`modp`, convert(numtheory:-divisors(n), list), 10);
numboccur(1, D) + numboccur(9, D) - numboccur(3, D) - numboccur(7, D);
end proc:
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MATHEMATICA
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a[n_] := With[{d = Mod[Divisors[n], 10]}, Count[d, 1|9] - Count[d, 3|7]];
a[n_] := DivisorSum[n, KroneckerSymbol[20, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)
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PROG
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(PARI) my(m = 20); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(20, d)); \\ Amiram Eldar, Nov 19 2023
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CROSSREFS
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KEYWORD
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nonn,easy,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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