A049060 a(n) = (-1)^omega(n)*Sum_{d|n} d*(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.

1, 1, 2, 5, 4, 2, 6, 13, 11, 4, 10, 10, 12, 6, 8, 29, 16, 11, 18, 20, 12, 10, 22, 26, 29, 12, 38, 30, 28, 8, 30, 61, 20, 16, 24, 55, 36, 18, 24, 52, 40, 12, 42, 50, 44, 22, 46, 58, 55, 29, 32, 60, 52, 38, 40, 78, 36, 28, 58, 40, 60, 30, 66, 125, 48, 20, 66, 80, 44, 24, 70

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice
• mult

Comments

Might be called (-1)sigma(n). If x = Product p_i^r_i, then (-1)sigma(x) = Product (-1 + Sum p_i^s_i, s_i = 1 to r_i) = Product ((p_i^(r_i+1)-1)/(p_i-1)-2), with (-1)sigma(1) = 1. - Yasutoshi Kohmoto, May 23 2005

Links

• R. J. Mathar, Table of n, a(n) for n = 1..100000

Crossrefs

Used in A049057, A049058, A049059.

Cf. A000203, A001221, A057723, A060640, A126602, A126690.

Programs

Maple

A049060 := proc(n) local it, ans, i, j; it := ifactors(n): ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(-1+sum(ifactors(n)[2][i][1]^j, j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end: [seq(A049060(i),i=1..n)];

Mathematica

a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ ((#[[1]]^(#[[2]] + 1) - 2*#[[1]] + 1)/(#[[1]] - 1) & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012 *)

PARI

A049060(n)={ local(i,resul,rmax,p) ; if(n==1, return(1) ) ; i=factor(n) ; rmax=matsize(i)[1] ; resul=1 ; for(r=1,rmax, p=0 ; for(j=1,i[r,2], p += i[r,1]^j ; ) ; resul *= p-1 ; ) ; return(resul) ; } { for(n=1,40, print(n," ",A049060(n)) ) ; } \\ R. J. Mathar, Oct 12 2006

PARI

apply( A049060(n)=vecprod([(f[1]^(f[2]+1)-1)\(f[1]-1)-2 | f<-factor(n)~]), [1..99]) \\ M. F. Hasler, Sep 21 2022

Python

from math import prod
from sympy import factorint
def A049060(n): return prod((p**(e+1)-2*p+1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Sep 13 2021

Formula

a(n) = Sum_{d|n} d*(-1)^A001221(d).

Multiplicative with a(p^e) = (p^(e+1)-2*p+1)/(p-1).

Simpler: a(p^e) = (p^(e+1)-1)/(p-1)-2. - M. F. Hasler, Sep 21 2022

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 2/p^2 + 2/p^3) = 0.4478559359... . - Amiram Eldar, Oct 25 2022

Extensions

More terms from James Sellers, May 03 2000

Better description from Vladeta Jovovic, Apr 06 2002