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A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).

%I A326689 #27 Feb 05 2024 01:01:23
%S A326689 -1,0,0,1,0,2,0,3,2,3,0,3,0,4,7,7,0,7,0,13,3,6,0,19,4,7,8,17,0,1,0,15,
%T A326689 13,9,11,29,0,10,5,27,0,20,0,25,23,12,0,13,6,17,19,29,0,22,3,5,7,15,0,
%U A326689 61,0,16,29,31,17,10,0,37,25,29,0,59,0,19,13,41
%N A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).
%C A326689 See Comments on denominators in A326690.
%H A326689 Antti Karttunen, <a href="/A326689/b326689.txt">Table of n, a(n) for n = 1..16384</a>
%H A326689 Antti Karttunen, <a href="/A326689/a326689.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A326689 Wikipedia, <a href="https://en.wikipedia.org/wiki/Giuga_number">Giuga number</a>
%F A326689 a(p) = 0 if p is a prime.
%F A326689 a(g) = 1 if g is a known Giuga number (see my 2nd comment in A007850).
%e A326689 -1/1, 0/1, 0/1, 1/4, 0/1, 2/3, 0/1, 3/8, 2/9, 3/5, 0/1, 3/4, 0/1, 4/7, 7/15, 7/16, 0/1, 7/9, 0/1, 13/20, 3/7, 6/11, 0/1, 19/24, 4/25, 7/13, 8/27, 17/28, 0/1, 1/1
%t A326689 PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
%t A326689 g[n_] := Numerator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
%t A326689 Table[ g[n], {n, 100}]
%o A326689 (PARI) a(n) = numerator(sumdiv(n, d, isprime(d)/d) - 1/n); \\ _Michel Marcus_, Jul 19 2019
%Y A326689 Denominators are A326690. Cf. also A007850, A309132, A309235, A309378.
%Y A326689 Cf. A028235.
%K A326689 sign,frac
%O A326689 1,6
%A A326689 _Jonathan Sondow_, Jul 18 2019