A281785 a(n) is multiplicative with a(2^e) = 1, a(3^e) = -8 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

1, 1, -8, 1, 6, -8, 8, 1, -8, 6, 12, -8, 14, 8, -48, 1, 18, -8, 20, 6, -64, 12, 24, -8, 31, 14, -8, 8, 30, -48, 32, 1, -96, 18, 48, -8, 38, 20, -112, 6, 42, -64, 44, 12, -48, 24, 48, -8, 57, 31, -144, 14, 54, -8, 72, 8, -160, 30, 60, -48, 62, 32, -64, 1, 84

Offset

1,3

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

Expansion of (a(x) * b(x^2) + a(x^2) * b(x) - 2) / 3 in powers of x where a(), b() are cubic AGM functions.

Expansion of (3 * b(x^3) * b(x^6) - b(x) * b(x^2) - 2) / 3 in powers of x where b() is a cubic AGM function.

3 * a(n) = A281786(n) if n>0. a(2*n) = a(n). a(3*n) = -8 * A186099(n).

Dirichlet g.f.: zeta(s) * zeta(s-1) * (1-2^(1-s)) * (1-3^(1-s)) * (1-3^(2-s)). - Amiram Eldar, Oct 24 2023

Example

G.f. = x + x^2 - 8*x^3 + x^4 + 6*x^5 - 8*x^6 + 8*x^7 + x^8 - 8*x^9 + 6*x^10 + ...

Maple

f:= n -> mul(piecewise( t[1] = 2, 1, t[1] = 3, -8, (t[1]^(t[2]+1)-1)/(t[1]-1)), t = ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Feb 01 2017

Mathematica

a[ n_] := If[ n < 1, 0, If[ Divisible[n, 3], -8, 1] DivisorSigma[ 1, n / (2^IntegerExponent[n, 2] 3^IntegerExponent[n, 3])]];
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, 1, # == 3, -8, True, (#^(#2+1) - 1) / (# - 1)] & @@@ FactorInteger@n)];

Prog

(PARI) {a(n) = if( n<1, 0, if( n%3, 1, -8) * sigma(n / (2^valuation(n, 2) * 3^valuation(n, 3))))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*(d%2)) - if( n%3==0, 12 * sumdiv(n/3, d, d*(d%2))) + if( n%9==0, 27 * sumdiv(n/9, d, d*(d%2))))};

Crossrefs

Cf. A186099, A281786.

Sequence in context: A007404 A299627 A157697 * A240982 A258146 A182551

Adjacent sequences: A281782 A281783 A281784 * A281786 A281787 A281788

Keyword

mult,sign

Author

Michael Somos, Jan 30 2017

Status

approved