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A049060
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a(n) = (-1)^omega(n)*Sum_{d|n} d*(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.
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27
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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
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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R. J. Mathar, Table of n, a(n) for n = 1..100000
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FORMULA
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a(n) = Sum_{d|n} d*(-1)^A001221(d).
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MAPLE
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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)];
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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Used in A049057, A049058, A049059. Cf. A000203, A057723, A060640, A126602, A126690.
Sequence in context: A152669 A324051 A307037 * A092462 A256357 A160826
Adjacent sequences: A049057 A049058 A049059 * A049061 A049062 A049063
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KEYWORD
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easy,nonn,nice,mult
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers, May 03 2000
Better description from Vladeta Jovovic, Apr 06 2002
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STATUS
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approved
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