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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A049236 A244259 A094840 * A237442 A277070 A139355

Adjacent sequences:  A035215 A035216 A035217 * A035219 A035220 A035221

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Antti Karttunen, Oct 03 2018

STATUS

approved

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Last modified March 26 16:58 EDT 2019. Contains 321511 sequences. (Running on oeis4.)