A393006 The number of positive integers below the (n+1)-st primorial number that are the product of exactly n distinct primes.

1, 4, 12, 32, 101, 264, 715, 1624, 3712, 9726, 21834, 53792, 128199, 267188, 539652, 1153300, 2579888, 5149067, 10688991, 21413547, 38696082, 72239075, 130346733, 241575166, 484716655, 939025292, 1666576578, 2860265647, 4508364585, 6875274627, 12911499333, 23435167650, 42980412215

Offset

0,2

Comments

The number of terms in A116998 that have exactly n distinct prime factors.

A116998 is the sequence of integers k for which the number of distinct prime factors, A001221(k), and the number of unitary divisors, A034444(k) = 2^A001221(k), are nondecreasing. Note that A001221(k) increases by 1 and A034444(k) doubles, whenever k is a primorial number (A002110). A116998 is analogous to Ramanujan's largely composite numbers (A067128), while A002110 is analogous to highly composite numbers (A002182), and this sequence is analogous to A308530.

Links

Table of n, a(n) for n=0..32.

Amiram Eldar, Visualization of the terms a(0)-a(3).

Example

  n | a(n) | primorial(n+1) | integers k < primorial(n+1) with omega(k) = n
  --+------+----------------+----------------------------------------------
  0 | 1    | 2              | 1
  1 | 4    | 6              | 2, 3, 4, 5
  2 | 12   | 30             | 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28

Mathematica

a[n_] := Module[{p2, maxp, pr, f}, p2 = Product[Prime[i], {i, 1, n + 1}]; maxp = Prime[n]*Prime[n+1]; pr = Prime[Range[PrimePi[maxp]]]; f[e_, lim_, ind_, v_] := f[e, lim, ind, v] = Module[{c = 0, i = ind, p, temp, minr}, If[e == 1, If[Floor[lim/v] < pr[[ind]], 0, PrimePi[Floor[lim/v]] - ind + 1], While[i <= Length[pr] && v * (p = pr[[i]]) * (minr = p^(e-1)) <= lim, temp = v*p; While[temp * minr <= lim, c += f[e-1, lim, i+1, temp]; temp *= p]; i++]; c]]; f[n, p2, 1, 1] + 1]; Array[a, 15, 0]

Prog

(PARI) a(n) = {my(p1 = vecprod(primes(n)), p2 = p1 * prime(n+1), c = 1); for(k = p1+1, p2-1, if(omega(k) == n, c++)); c; }

Crossrefs

The unitary analog of A308530.

Cf. A001221, A002110, A002182, A034444, A067128, A116998, A393007 (infinitary analog).

Sequence in context: A120369 A001665 A066536 * A265212 A295500 A168078

Adjacent sequences: A393003 A393004 A393005 * A393007 A393008 A393009

Keyword

nonn

Author

Amiram Eldar, Jan 30 2026

Status

approved