A125131 Product 1-p, where p ranges over the prime factors of n with multiplicity.
1, -1, -2, 1, -4, 2, -6, -1, 4, 4, -10, -2, -12, 6, 8, 1, -16, -4, -18, -4, 12, 10, -22, 2, 16, 12, -8, -6, -28, -8, -30, -1, 20, 16, 24, 4, -36, 18, 24, 4, -40, -12, -42, -10, -16, 22, -46, -2, 36, -16, 32, -12, -52, 8, 40, 6, 36, 28, -58, 8, -60, 30, -24, 1, 48, -20, -66, -16, 44, -24, -70, -4, -72, 36, -32
Offset: 1
Examples
a(120) = -8 because the prime factorization of 120 is 2*2*2*3*5, so f(x)=(x-2)(x-2)(x-2)(x-3)(x-5) and f(1)=(-1)*(-1)*(-1)*(-2)*(-4)= -8.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
- Index to divisibility sequences
Programs
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Maple
a:= n-> mul((1-i[1])^i[2], i=ifactors(n)[2]): seq(a(n), n=1..80); # Alois P. Heinz, Jun 28 2015
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Mathematica
a[1] = 1; a[n_] := Times @@ (-Flatten[Table[ #1, {#2}] & @@@ FactorInteger@n] + 1); Array[a, 80] (* Robert G. Wilson v, Jan 10 2007; corrected by Michael Shamos, Aug 12 2023 and May 27 2025 *) -
PARI
a(n)=my(f=factor(n)); prod(i=1,#f~,(1-f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015
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PARI
for(n=1, 100, print1(direuler(p=2, n, 1/(1 + p*X - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020
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R
f=polyroot(factor(x)); f(1)
Formula
a(n) = f(1) where f is the monic polynomial whose zeros are the prime factors of n with multiplicity.
a(p) = 1-p for any prime number p.
Completely multiplicative with a(p) = 1-p. - Franklin T. Adams-Watters, Jan 17 2007
a(n) = f(1), where f(x)=(x-p_1)(x-p_2)...(x-p_m), where { p_1,p_2,...p_m } are the prime factors of n with multiplicity.
Dirichlet g.f.: Product_{primes p} 1/(1 + p^(1-s) - p^(-s)). - Vaclav Kotesovec, Jun 14 2020
Extensions
Edited by Franklin T. Adams-Watters, Jan 17 2007