A393006 - OEIS

A393006

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

%I #12 Feb 01 2026 05:20:01

%S 1,4,12,32,101,264,715,1624,3712,9726,21834,53792,128199,267188,

%T 539652,1153300,2579888,5149067,10688991,21413547,38696082,72239075,

%U 130346733,241575166,484716655,939025292,1666576578,2860265647,4508364585,6875274627,12911499333,23435167650,42980412215

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

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

%C 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.

%H Amiram Eldar, <a href="/A393006/a393006.png">Visualization of the terms a(0)-a(3)</a>.

%e n | a(n) | primorial(n+1) | integers k < primorial(n+1) with omega(k) = n

%e --+------+----------------+----------------------------------------------

%e 0 | 1 | 2 | 1

%e 1 | 4 | 6 | 2, 3, 4, 5

%e 2 | 12 | 30 | 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28

%t 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]

%o (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;}

%Y The unitary analog of A308530.

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

%K nonn

%O 0,2

%A _Amiram Eldar_, Jan 30 2026