A035202 - OEIS

%I #25 Nov 19 2023 01:27:26

%S 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,

%T 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,

%U 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

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 20.

%C 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

%H Amiram Eldar, <a href="/A035202/b035202.txt">Table of n, a(n) for n = 1..10000</a>

%H MathNerds, <a href="https://web.archive.org/web/20130123081526/http://www.mathnerds.com/best/DivisorExcess/index.aspx">An Excess of Divisors</a>. [Wayback Machine link]

%F From _Amiram Eldar_, Nov 19 2023: (Start)

%F a(n) = Sum_{d|n} Kronecker(20, d).

%F 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).

%F 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)

%p a:= proc(n) local D,d; D:= map(`modp`,convert(numtheory:-divisors(n),list),10);

%p       numboccur(1,D) + numboccur(9,D) - numboccur(3,D) - numboccur(7,D);

%p end proc:

%p seq(a(n),n=1..1000); # _Robert Israel_, Sep 22 2014

%t a[n_] := With[{d = Mod[Divisors[n], 10]}, Count[d, 1|9] - Count[d, 3|7]];

%t Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, May 15 2023 *)

%t a[n_] := DivisorSum[n, KroneckerSymbol[20, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2023 *)

%o (PARI) my(m = 20); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(20, d)); \\ _Amiram Eldar_, Nov 19 2023

%Y Cf. A083911, A083913, A083917, A083919.

%Y Cf. A001622, A003631, A045468.

%K nonn,easy,mult

%O 1,11

%A _N. J. A. Sloane_

%E More terms from _Henry Bottomley_, Jul 08 2003