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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. 28
 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 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

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Last modified December 2 06:18 EST 2023. Contains 367509 sequences. (Running on oeis4.)
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