A035218 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 36.

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 4, 2, 1, 2, 2, 3, 2, 4, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 3, 2, 3, 2, 2, 2, 2, 4

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

Comments

a(n) is the number of factors (over Q) of the polynomial x^(2n) - x^n + 1. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003

This sequence is multiplicative. Just as (A001227)(n) is the number of ways to write n as differences of 3-gonal numbers, this sequence is the number of ways to write n as difference of (-1)-gonal numbers. If p_e(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-1. - Volker Schmitt (clamsi(AT)gmx.net), Oct 11 2004

a(n) is the number of divisors of n not divisible by 2 or 3. For example, a(36)=1 because 1 is the only such divisor of 36. a(10) = 2 because we count the divisors 1 and 5. - Geoffrey Critzer, Feb 15 2015

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A035191, A000005, A001227, A279060, A319995, A320015, A001620.

Programs

Maple

res:=1; ifac:=op(ifactors(i))[2]; for pfac in ifac do; if pfac[1]>3 then res:=res*(pfac[2]+1); a(n):=res;

Mathematica

nn = 81; f[list_, i] := list[[i]]; a = Prepend[Drop[Table[Boole[Min[FactorInteger[n][[All, 1]]] > 3], {n, 1, nn}], 1], 1]; b = Table[1, {nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 15 2015 *)
f[p_, e_] := If[p >= 5, e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

PARI

m=36; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

PARI

a(n) = sumdiv(n, d, (d % 2) && (d % 3)); \\ Michel Marcus, Feb 16 2015

Formula

a(n) = d(6n) - d(3n) - d(2n) + d(n) where d() is the divisor function. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003

Multiplicative with a(2^e)=1, a(3^e)=1, a(p^e)=e+1 if p>3. Inverse Möbius transform is periodic with 1, 0, 0, 0, 1, 0. - Volker Schmitt (clamsi(AT)gmx.net), Oct 11 2004

Dirichlet g.f.: zeta(s)^2*(1 - 1/2^s)*(1 - 1/3^s). - Geoffrey Critzer, Feb 15 2015

From Antti Karttunen, Oct 03 2018: (Start)

a(n) = A279060(n) + A319995(n).

a(n) = A320015(n) + ch15(n), where ch15 is the characteristic function of numbers of the form +-1 mod 6, i.e., ch15(n) = A232991(n-1).

(End)

Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma + log(12)/2 - 1)/3, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 29 2019

Extensions

More terms from Antti Karttunen, Oct 03 2018