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

1,5

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.

More specifically, a(n) is the numerator of the proportion of numbers whose second smallest prime divisor is prime(n) in every interval of A002110(n) positive integers, where A002110(n) is the product of the first n primes. The number in each such interval is A078456(n-1), n > 1. Compare with column 2 of A281890. - Peter Munn, Mar 13 2026

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.

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).

For n > 1, a(n) = A078456(n-1)/gcd(A078456(n-1), A002110(n)). - Peter Munn, Mar 13 2026

Example

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.

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]

Crossrefs

Cf. A027748, A038110, A038111, A119288, A342480 (denominators).

See the comments for the relationships with A000961, A002110, A078456, A281890.

Sequence in context: A060958 A094892 A258915 * A165867 A058905 A345492

Adjacent sequences: A342476 A342477 A342478 * A342480 A342481 A342482

Keyword

nonn,easy,frac

Author

Amiram Eldar, Mar 13 2021

Status

approved