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

0, 1, 1, 1, 46, 44, 288, 33216, 613248, 151296, 391584768, 2383570944, 86830424064, 206470840320, 21270238986240, 987259950858240, 1262040231444480, 3022250536693923840, 3884253754215628800, 1102040800033347993600, 1892288242221318144000, 5616902226049109065728000

Offset: 1

Author

Amiram Eldar, Mar 13 2021

Keywords

• nonn
• easy
• frac

Comments

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.

Examples

			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, ...
a(1) = 0 since there are no numbers whose second smallest prime divisor is prime(1) = 2.
a(2)/A342480(2) = 1/6 since the numbers whose second smallest prime divisor is prime(2) = 3 are the positive multiples of 6.
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.
		

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..376
• Paul Erdős and Gérald Tenenbaum, Sur les densités de certaines suites d'entiers, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; alternative link.

Crossrefs

Cf. A000961, A038110, A038111, A119288, A342480 (denominators).

Programs

Mathematica

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]

Formula

a(n)/A342480(n) = (1/prime(n)) * Product_{q prime < prime(n)} (1 - 1/q) * Sum_{q prime < prime(n)} 1/(q-1).

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).