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%I #27 Aug 13 2023 08:49:04
%S 1,-1,-2,1,-4,2,-6,-1,4,4,-10,-2,-12,6,8,1,-16,-4,-18,-4,12,10,-22,2,
%T 16,12,-8,-6,-28,-8,-30,-1,20,16,24,4,-36,18,24,4,-40,-12,-42,-10,-16,
%U 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
%N Product 1-p, where p ranges over the prime factors of n with multiplicity.
%H Alois P. Heinz, <a href="/A125131/b125131.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = f(1) where f is the monic polynomial whose zeros are the prime factors of n with multiplicity.
%F a(p) = 1-p for any prime number p.
%F Completely multiplicative with a(p) = 1-p. - _Franklin T. Adams-Watters_, Jan 17 2007
%F 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.
%F a(n) = A003958(n) * A008836(n).
%F Dirichlet g.f.: Product_{primes p} 1/(1 + p^(1-s) - p^(-s)). - _Vaclav Kotesovec_, Jun 14 2020
%e 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.
%p a:= n-> mul((1-i[1])^i[2], i=ifactors(n)[2]):
%p seq(a(n), n=1..80); # _Alois P. Heinz_, Jun 28 2015
%t a[n_] := Times @@ (-Flatten[Table[ #1, {#2}] & @@@ FactorInteger@n] + 1);
%t Array[a, 80] (* _Robert G. Wilson v_, Jan 10 2007; corrected by _Michael Shamos_, Aug 12 2023 *)
%o (R) f=polyroot(factor(x)); f(1)
%o (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
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 + p*X - X))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020
%Y Cf. A003958, A008836.
%K easy,sign,mult
%O 1,3
%A Mitch Cervinka (puritan(AT)toast.net), Jan 10 2007
%E Edited by _Franklin T. Adams-Watters_, Jan 17 2007
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