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A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

%I A332880 #21 Nov 24 2022 04:20:57
%S A332880 1,3,4,3,6,2,8,3,4,9,12,2,14,12,8,3,18,2,20,9,32,18,24,2,6,21,4,12,30,
%T A332880 12,32,3,16,27,48,2,38,30,56,9,42,16,44,18,8,36,48,2,8,9,24,21,54,2,
%U A332880 72,12,80,45,60,12,62,48,32,3,84,24,68,27,32,72
%N A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).
%C A332880 Numerator of sum of reciprocals of squarefree divisors of n.
%C A332880 (6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - _Amiram Eldar_, Nov 24 2022
%H A332880 Antti Karttunen, <a href="/A332880/b332880.txt">Table of n, a(n) for n = 1..20000</a>
%F A332880 Numerators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
%F A332880 a(n) = numerator of Sum_{d|n} mu(d)^2/d.
%F A332880 a(n) = numerator of psi(n)/n.
%F A332880 a(p) = p + 1, where p is prime.
%F A332880 a(n) = A001615(n) / A306695(n) = A001615(n) / gcd(n, A001615(n)). - _Antti Karttunen_, Nov 15 2021
%F A332880 From _Amiram Eldar_, Nov 24 2022: (Start)
%F A332880 Asymptotic means:
%F A332880 Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).
%F A332880 Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)
%e A332880 1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...
%p A332880 a:= n-> numer(mul(1+1/i[1], i=ifactors(n)[2])):
%p A332880 seq(a(n), n=1..80);  # _Alois P. Heinz_, Feb 28 2020
%t A332880 Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
%t A332880 Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Numerator
%o A332880 (PARI)
%o A332880 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
%o A332880 A332880(n) = numerator(A001615(n)/n);
%Y A332880 Cf. A001615, A008683, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.
%Y A332880 Cf. also A348734, A348735.
%Y A332880 Cf. A059956, A065463, A082020, A094387, A339076.
%K A332880 nonn,frac
%O A332880 1,2
%A A332880 _Ilya Gutkovskiy_, Feb 28 2020