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

1,3

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

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

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 *)

Prog

(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

Crossrefs

Used in A049057, A049058, A049059.

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

Sequence in context: A152669 A324051 A307037 * A092462 A256357 A160826

Adjacent sequences: A049057 A049058 A049059 * A049061 A049062 A049063

Keyword

easy,nonn,nice,mult

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 03 2000

Better description from Vladeta Jovovic, Apr 06 2002

Status

approved