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A342479 a(n) is the numerator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

%I A342479 #14 Mar 14 2021 05:20:20
%S A342479 0,1,1,1,46,44,288,33216,613248,151296,391584768,2383570944,
%T A342479 86830424064,206470840320,21270238986240,987259950858240,
%U A342479 1262040231444480,3022250536693923840,3884253754215628800,1102040800033347993600,1892288242221318144000,5616902226049109065728000
%N A342479 a(n) is the numerator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).
%C A342479 The second smallest prime divisor of a number k is the second member in the ordered list of the distinct prime divisors of k. All the numbers that are not prime powers (A000961) have a second smallest prime divisor.
%D A342479 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.
%H A342479 Amiram Eldar, <a href="/A342479/b342479.txt">Table of n, a(n) for n = 1..376</a>
%H A342479 Paul Erdős and Gérald Tenenbaum, <a href="https://doi.org/10.1112/plms/s3-59.3.417">Sur les densités de certaines suites d'entiers</a>, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; <a href="https://users.renyi.hu/~p_erdos/1989-36.pdf">alternative link</a>.
%F A342479 a(n)/A342480(n) = (1/prime(n)) * Product_{q prime < prime(n)} (1 - 1/q) * Sum_{q prime < prime(n)} 1/(q-1).
%F A342479 Sum_{n>=1} a(n)/A342480(n) = 1 (since the asymptotic density of numbers without a second smallest prime divisor, i.e., the prime powers, is 0).
%e A342479 The fractions begin with 0, 1/6, 1/10, 1/15, 46/1155, 44/1365, 288/12155, 33216/1616615, 613248/37182145, 151296/11849255, 391584768/33426748355, ...
%e A342479 a(1) = 0 since there are no numbers whose second smallest prime divisor is prime(1) = 2.
%e A342479 a(2)/A342480(2) = 1/6 since the numbers whose second smallest prime divisor is prime(2) = 3 are the positive multiples of 6.
%e A342479 a(3)/A342480(3) = 1/10 since the numbers whose second smallest prime divisor is prime(3) = 5 are the numbers congruent to {10, 15, 20} (mod 30) whose density is 3/30 = 1/10.
%t A342479 f[n_] := Module[{p = Prime[n], q}, q = Select[Range[p - 1], PrimeQ]; Plus @@ (1/(q - 1))*Times @@ ((q - 1)/q)/p]; Numerator @ Array[f, 30]
%Y A342479 Cf. A000961, A038110, A038111, A119288, A342480 (denominators).
%K A342479 nonn,easy,frac
%O A342479 1,5
%A A342479 _Amiram Eldar_, Mar 13 2021