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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. 23
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 (list; graph; refs; listen; history; text; internal format)
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

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

LINKS

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

FORMULA

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

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-Fran├žois 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

CROSSREFS

Used in A049057, A049058, A049059. Cf. A000203, A057723, A060640, A126602, A126690.

Sequence in context: A177067 A055127 A152669 * 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

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)