A035202 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 20.
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
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- MathNerds, An Excess of Divisors. [Wayback Machine link]
Programs
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Maple
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: seq(a(n),n=1..1000); # Robert Israel, Sep 22 2014 -
Mathematica
a[n_] := With[{d = Mod[Divisors[n], 10]}, Count[d, 1|9] - Count[d, 3|7]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 15 2023 *) a[n_] := DivisorSum[n, KroneckerSymbol[20, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *) -
PARI
my(m = 20); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
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PARI
a(n) = sumdiv(n, d, kronecker(20, d)); \\ Amiram Eldar, Nov 19 2023
Formula
From Amiram Eldar, Nov 19 2023: (Start)
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)
Extensions
More terms from Henry Bottomley, Jul 08 2003
Comments