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A364813 a(n) = Product_{k=2..n} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise ord(n, k) = e where e is such that k^e divides n but k^(e + 1) does not.

%I A364813 #26 Apr 02 2025 04:11:33
%S A364813 1,2,3,16,5,36,7,256,81,100,11,3456,13,196,225,32768,17,17496,19,
%T A364813 16000,441,484,23,1327104,625,676,6561,43904,29,810000,31,2097152,
%U A364813 1089,1156,1225,362797056,37,1444,1521,10240000,41,3111696,43,170368,273375,2116,47,8153726976,2401,625000
%N A364813 a(n) = Product_{k=2..n} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise ord(n, k) = e where e is such that k^e divides n but k^(e + 1) does not.
%C A364813 a(n) is divisible by n and a(p) = p if p is prime. More general, if the base of the factors of the product is restricted to prime numbers then the positive integers are generated according to the fundamental theorem of arithmetic. - _Peter Luschny_, Apr 01 2025
%H A364813 Michel Marcus, <a href="/A364813/b364813.txt">Table of n, a(n) for n = 1..10000</a>
%H A364813 Jeffrey C. Lagarias and Wijit Yangjit, <a href="https://arxiv.org/abs/2310.12949">The factorial function and generalizations, extended</a>, arXiv:2310.12949 [math.NT], 2023. See Section 7.2 pp. 20-21 and Table 1 p. 29.
%H A364813 Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_valuation">p-adic valuation</a>.
%F A364813 a(n) = Product_{b=2..n} b^d(n, b) where d(n, b) = gamma(n, b) - gamma(n-1, b) and gamma(n, b) = Sum_{i>=1} floor(n/b^i).
%p A364813 with(padic): a := n -> local k; mul(k^ordp(n, k), k = 2.. n): seq(a(n), n = 1..50); # _Peter Luschny_, Apr 01 2025
%t A364813 Table[Product[k^IntegerExponent[n, k], {k, 2, n}], {n, 1, 50}] (* _Peter Luschny_, Apr 01 2025 *)
%o A364813 (PARI) f(n, b) = sum(i=1, logint(n, b), n\b^i);
%o A364813 d(n,b) = f(n,b)-f(n-1,b);
%o A364813 a(n) = prod(b=2, n, b^d(n,b));
%Y A364813 Cf. A363838 (also uses gamma), A000027, A005451, A381885 (a(n)/n).
%K A364813 nonn
%O A364813 1,2
%A A364813 _Michel Marcus_, Oct 21 2023
%E A364813 More explicit name from _Peter Luschny_, Apr 01 2025