A309912 a(n) = Product_{p prime, p <= n} floor(n/p).

1, 1, 1, 1, 2, 2, 6, 6, 8, 12, 30, 30, 48, 48, 112, 210, 240, 240, 324, 324, 480, 840, 1848, 1848, 2304, 2880, 6240, 7020, 10080, 10080, 14400, 14400, 15360, 25344, 53856, 78540, 90720, 90720, 191520, 311220, 374400, 374400, 508032, 508032, 709632, 855360, 1788480, 1788480

Offset: 0

Author

Ilya Gutkovskiy, Aug 22 2019

Keywords

• nonn

Comments

Product of exponents of prime factorization of A048803 (squarefree factorials).

Examples

			A048803(14) = 1816214400 = 2^7 * 3^4 * 5^2 * 7^2 * 11 * 13 so a(14) = 7 * 4 * 2 * 2 * 1 * 1 = 112.
		

Crossrefs

Cf. A000040, A000720, A005361, A010786, A013939, A048803, A135291.

Programs

Maple

a:= n-> mul(floor(n/p), p=select(isprime, [$2..n])):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 23 2019

Mathematica

Table[Product[Floor[n/Prime[k]], {k, 1, PrimePi[n]}], {n, 0, 47}]

Python

from math import prod
from sympy import primerange
def A309912(n): return prod(n//p for p in primerange(n)) # Chai Wah Wu, Jun 02 2025

Formula

a(n) = Product_{k=1..A000720(n)} floor(n/A000040(k)).

a(n) = A005361(A048803(n)).