This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363596 #11 Aug 17 2025 14:13:26
%S A363596 1,1,2,1,6,2,12,3,10,2,12,2,420,60,24,3,90,10,420,42,660,60,360,30,
%T A363596 3276,252,56,4,120,8,3696,231,3570,210,36,2,103740,5460,840,42,13860,
%U A363596 660,27720,1260,19320,840,5040,210,198900,7956,10296,396,11880,440,6384,228
%N A363596 a(n) = (Product_{k=1..pi(n+1)} prime(k)^floor(n/(prime(k)-1) ) )/(n+1)!.
%C A363596 Motivated by Proposition 3.2, p. 10 of the Bedhouche-Farhi paper.
%C A363596 Observations regarding prime power decomposition of terms in a(0..20737):
%C A363596 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.
%C A363596 a(34) = 36 is the only term in A286708 (more specifically, in A303606).
%C A363596 a(35) = 2 is the last prime term.
%C A363596 a(29) = 8 is the only composite prime power.
%C A363596 a(190) = 221760 is the last term in A002182, but a(61) = a(102) = 720720 is the largest.
%C A363596 a(191) = 2310 is the last primorial term.
%C A363596 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).
%C A363596 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).
%H A363596 Michael De Vlieger, <a href="/A363596/b363596.txt">Table of n, a(n) for n = 0..10000</a>
%H A363596 Abdelmalek Bedhouche and Bakir Farhi, <a href="https://arxiv.org/abs/2207.07957">On some products taken over the prime numbers</a>, arXiv:2207.07957 [math.NT], 2022. See p. 10.
%H A363596 Michael De Vlieger, <a href="/A363596/a363596.png">Log log scatterplot of a(n+1)</a>, n = 0..10^4.
%H A363596 Michael De Vlieger, <a href="/A363596/a363596_1.png">Plot p(k)^e(k) | a(n) at (x, y) = (n, k)</a>, 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.
%F A363596 a(n) = A091137(n)/(n+1)!.
%e A363596 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).
%e A363596 n A067255(b(n)) (n+1)!*a(n) k! * m
%e A363596 ---------------------------------------
%e A363596 0 0 1! * 1 1! * 1
%e A363596 1 1 2! * 1 2! * 1
%e A363596 2 2.1 3! * 2 3! * 2
%e A363596 3 3.1 4! * 1 4! * 1
%e A363596 4 4.2.1 5! * 6 6! * 1
%e A363596 5 5.2.1 6! * 2 6! * 2
%e A363596 6 6.3.1.1 7! * 12 7! * 12
%e A363596 7 7.3.1.1 8! * 3 8! * 3
%e A363596 8 8.4.2.1 9! * 10 10! * 1
%e A363596 9 9.4.2.1 10! * 2 10! * 2
%e A363596 10 10.5.2.1.1 11! * 12 12! * 1
%e A363596 11 11.5.2.1.1 12! * 2 12! * 2
%e A363596 12 12.6.3.2.1.1 13! * 420 15! * 2
%e A363596 13 13.6.3.2.1.1 14! * 60 15! * 4
%e A363596 14 14.7.3.2.1.1 15! * 24 15! * 24
%e A363596 15 15.7.3.2.1.1 16! * 3 16! * 3
%e A363596 16 16.8.4.2.1.1.1 17! * 90 18! * 5
%e A363596 ...
%t A363596 Table[j = 1; ( Times @@ Reap[While[Sow[#^Floor[n/(# - 1)]] &[Prime[j]] > 1, j++]][[-1, 1]] )/Factorial[n + 1], {n, 0, 60}]
%o A363596 (Python)
%o A363596 from math import prod, factorial
%o A363596 from sympy import sieve
%o A363596 def A363596(n: int) -> int:
%o A363596 numer = prod(p ** (n // (p - 1)) for p in sieve.primerange(2, n + 2))
%o A363596 return numer // factorial(n + 1)
%o A363596 print([A363596(n) for n in range(56)]) # _Peter Luschny_, Aug 17 2025
%Y A363596 Cf. A000142, A000720, A091137, A361098.
%K A363596 nonn,easy
%O A363596 0,3
%A A363596 _Michael De Vlieger_, Aug 03 2023