A363596 a(n) = (Product_{k=1..pi(n+1)} prime(k)^floor(n/(prime(k)-1) ) )/(n+1)!.

1, 1, 2, 1, 6, 2, 12, 3, 10, 2, 12, 2, 420, 60, 24, 3, 90, 10, 420, 42, 660, 60, 360, 30, 3276, 252, 56, 4, 120, 8, 3696, 231, 3570, 210, 36, 2, 103740, 5460, 840, 42, 13860, 660, 27720, 1260, 19320, 840, 5040, 210, 198900, 7956, 10296, 396, 11880, 440, 6384, 228

Offset: 0

Author

Michael De Vlieger, Aug 03 2023

Keywords

• nonn
• easy

Comments

Motivated by Proposition 3.2, p. 10 of the Bedhouche-Farhi paper.

Observations regarding prime power decomposition of terms in a(0..20737):

For n > 300, most terms are in A361098 but not in A286708. A303606 is a subset of A286708, which is a subset of A361098, which in turn is a subset of A126706, numbers that are neither prime powers nor squarefree.

a(34) = 36 is the only term in A286708 (more specifically, in A303606).

a(35) = 2 is the last prime term.

a(29) = 8 is the only composite prime power.

a(190) = 221760 is the last term in A002182, but a(61) = a(102) = 720720 is the largest.

a(191) = 2310 is the last primorial term.

a(1055) = 2207550414530882190 is the last squarefree term. If there are further squarefree terms a(n), n is likely to belong to -1 (mod 24).

a(7055) = 1733187515208453605856007490304335826298500960 is the last term that is not in A361098. a(n) not in A361098 is likely to belong to -1 (mod 24).

Examples

			The table below relates b(n) = A091137(n) to a(n), with (n+1)!*a(n) = k!*m = b(n), where k! is the largest factorial that divides b(n).
 n  A067255(b(n)) (n+1)!*a(n)   k! * m
---------------------------------------
 0  0                1! * 1     1! * 1
 1  1                2! * 1     2! * 1
 2  2.1              3! * 2     3! * 2
 3  3.1              4! * 1     4! * 1
 4  4.2.1            5! * 6     6! * 1
 5  5.2.1            6! * 2     6! * 2
 6  6.3.1.1          7! * 12    7! * 12
 7  7.3.1.1          8! * 3     8! * 3
 8  8.4.2.1          9! * 10   10! * 1
 9  9.4.2.1         10! * 2    10! * 2
10  10.5.2.1.1      11! * 12   12! * 1
11  11.5.2.1.1      12! * 2    12! * 2
12  12.6.3.2.1.1    13! * 420  15! * 2
13  13.6.3.2.1.1    14! * 60   15! * 4
14  14.7.3.2.1.1    15! * 24   15! * 24
15  15.7.3.2.1.1    16! * 3    16! * 3
16  16.8.4.2.1.1.1  17! * 90   18! * 5
...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See p. 10.
• Michael De Vlieger, Log log scatterplot of a(n+1), n = 0..10^4.
• Michael De Vlieger, Plot p(k)^e(k) | a(n) at (x, y) = (n, k), n = 0..2^11, with a color function representing e(k), where black = 1, red = 2, and the largest exponent in the dataset shown in magenta. The bar at bottom shows the number 1 in black, primes in red, composite prime powers in gold, squarefree terms in green, and terms that are neither squarefree nor prime powers in blue.

Crossrefs

Cf. A000142, A000720, A091137, A361098.

Programs

Mathematica

Table[j = 1; ( Times @@ Reap[While[Sow[#^Floor[n/(# - 1)]] &[Prime[j]] > 1, j++]][[-1, 1]] )/Factorial[n + 1], {n, 0, 60}]

Python

from math import prod, factorial
from sympy import sieve
def A363596(n: int) -> int:
    numer = prod(p ** (n // (p - 1)) for p in sieve.primerange(2, n + 2))
    return numer // factorial(n + 1)
print([A363596(n) for n in range(56)])  # Peter Luschny, Aug 17 2025

Formula

a(n) = A091137(n)/(n+1)!.