A378721 a(n) is the denominator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).

1, 1, 30, 30, 165, 15015, 36465, 62985, 7436429, 11849255, 73465381, 33426748355, 50708377254535, 436092044389001, 1863302371480277, 1086305282573001491, 64092011671807087969, 3909612711980232366109, 8449808119441147371913, 18598027670889965365580513, 3543193335582015099413

Offset: 1

Author

Robert G. Wilson v and Amiram Eldar, Dec 05 2024

Keywords

• nonn
• easy
• frac

Comments

See A378720 for more details.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..365

Crossrefs

Cf. A000040, A038110, A038111, A342479, A342480, A378720 (numerators).

Programs

Mathematica

a[n_] := Block[{p, q = Prime@ Range@ n}, p = Fold[Times, 1, q]; q = Most@ q; Plus @@ Times @@@ Subsets[q -1, {n -3}]/p]; a[1] = 0; Denominator@ Array[a, 21]

PARI

a(n) = {my(v = primes(n), q = vecextract(apply(x -> x-1, v),"^-1"), p = vecprod(v), prd = vecprod(q)/p, sm = 0, sb); forsubset([#q, 2], s, sb = vecextract(q, s); sm += 1/vecprod(sb)); denominator(prd * sm);}