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 A125131 Product 1-p, where p ranges over the prime factors of n with multiplicity. 4

%I

%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.

%C f(1), where f is the monic polynomial whose zeros are the prime factors of n with multiplicity.

%C a(p) = 1-p for any prime number p.

%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 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).

%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)^i, i=ifactors(n)):

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Jun 28 2015

%t f[n_] := Times @@ (-Flatten[Table[ #1, {#2}] & @@@ FactorInteger@n] + 1); Array[g, 80] (* _Robert G. Wilson v_, Jan 10 2007 *)

%o (?) 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

%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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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)