A092673 a(n) = moebius(n) - moebius(n/2) where moebius(n) is zero if n is not an integer.

1, -2, -1, 1, -1, 2, -1, 0, 0, 2, -1, -1, -1, 2, 1, 0, -1, 0, -1, -1, 1, 2, -1, 0, 0, 2, 0, -1, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, 1, 0, -1, -2, -1, -1, 0, 2, -1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 1, 2, -1, 1, -1, 2, 0, 0, 1, -2, -1, -1, 1, -2, -1, 0, -1, 2, 0, -1, 1, -2, -1, 0, 0, 2, -1, 1, 1, 2, 1, 0, -1, 0, 1, -1, 1, 2, 1, 0, -1, 0, 0, 0, -1, -2, -1, 0, -1, 2

Offset: 1

Author

Jon Perry, Mar 02 2004

Keywords

• sign
• mult

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Setting x=1 gives us phi(n) (A000010). Setting x=0 gives A092674.

Apparently the Dirichlet inverse of A001511. - R. J. Mathar, Dec 22 2010

Given A = A115359 as an infinite lower triangular matrix and B = the Mobius sequence as a vector, A092673 = A*B. - Gary W. Adamson, Mar 14 2011

Empirical: Letting M(n) denote the n X n matrix whereby the (i,j)-entry of M(n) is Sum_{k=1..j} floor(i/k), we have that a(n) is the (n,1)-entry of the inverse of M(n). - John M. Campbell, Aug 30 2017

John Campbell's statement is proved at the Mathematics Stack Exchange link. - Sungjin Kim, Jul 17 2019

Examples

			The first few s[n] are:
x, -2*x + 3, -x + 3, x + 1, -x + 5, 2*x, -x + 7, 4, 6, 2*x + 2, -x + 11, -x + 5, -x + 13, 2*x + 4, x + 7, 8, -x + 17, 6, -x + 19, -x + 9, x + 11, 2*x + 8, -x + 23, 8, 20, 2*x + 10, 18, -x + 13, -x + 29, -2*x + 10, -x + 31, 16, x + 19.
x - 2*x^2 - x^3 + x^4 - x^5 + 2*x^6 - x^7 + 2*x^10 - x^11 +...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Mathematics Stack Exchange, What is the inverse of ... ?.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A008683 (moebius(n)), A092149 (partial sums), A092674, A115359.

Cf. A000010, A000122, A000700, A010054, A092673, A115359, A121373.

Programs

Maple

A092673:= proc(n) if n::odd then numtheory:-mobius(n) else numtheory:-mobius(n) - numtheory:-mobius(n/2) fi end proc:
map(A092673, [$1..100]); # Robert Israel, Dec 31 2015

Mathematica

f[n_] := MoebiusMu[n] - If[OddQ@n, 0, MoebiusMu[n/2]]; Array[f, 105] (* Robert G. Wilson v *)

PARI

s=vector(2000); t(n)=binomial(n+1,2); s[1]=x; for(i=2,2000, s[i]=t(i)-sum(j=1,i-1, s[j]*floor(i/j))); for(i=1,2000,print1(","polcoeff(s[i],1)))

PARI

{a(n) = if( n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2)))} /* Michael Somos, Mar 26 2007 */

PARI

{a(n) = local(A, B, m); if( n<1, 0, A = x * O(x^n); B = 1 + x + A; for( k=1, n, B *= eta(x^k + A)^( m = polcoeff(B, k))); m)} /* Michael Somos, Mar 26 2007 */

PARI

a(n)=my(o=valuation(n%8,2)); if(o==0, moebius(n), if(o==1, 2*moebius(n), if(o==2, moebius(n/4), 0))) \\ Charles R Greathouse IV, Feb 07 2013

Python

from sympy import mobius
def A092673(n): return mobius(n)-(0 if n&1 else mobius(n>>1)) # Chai Wah Wu, Jul 13 2022

Formula

Let t(n) = binomial(n+1,2); s[1]=x; for i >= 2, s[i] = t(i)-Sum_{j=1..i-1} s[j]*floor(i/j); a(n) = coefficient of x in s[n]. - Jon Perry

a(n) is multiplicative with a(2)= -2, a(4)= 1, a(2^e)= 0 if e>2. a(p)= -1, a(p^e)= 0 if e>1, p>2. - Michael Somos, Mar 26 2007

a(8*n)= 0. a(2*n + 1) = moebius(2*n + 1). a(2*n) = moebius(2*n) - moebius(n). - Michael Somos, Mar 26 2007

|a(n)| <= 2.

1 / (1 + x) = Product_{k>0} f(-x^k)^a(k) where f() is a Ramanujan theta function. - Michael Somos, Mar 26 2007

Dirichlet g.f.: (1-2^(-s))/zeta(s). - Ralf Stephan, Mar 24 2015

G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} A(x^k). - Seiichi Manyama, Mar 31 2023

Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 9/Pi^2 = 0.9118906... . - Amiram Eldar, Jan 19 2024