OFFSET
0,3
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(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).
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.
FORMULA
a(n) = A091137(n)/(n+1)!.
EXAMPLE
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
...
MATHEMATICA
Table[j = 1; ( Times @@ Reap[While[Sow[#^Floor[n/(# - 1)]] &[Prime[j]] > 1, j++]][[-1, 1]] )/Factorial[n + 1], {n, 0, 60}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Aug 03 2023
STATUS
approved