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.

1, 2, 3, 16, 5, 36, 7, 256, 81, 100, 11, 3456, 13, 196, 225, 32768, 17, 17496, 19, 16000, 441, 484, 23, 1327104, 625, 676, 6561, 43904, 29, 810000, 31, 2097152, 1089, 1156, 1225, 362797056, 37, 1444, 1521, 10240000, 41, 3111696, 43, 170368, 273375, 2116, 47, 8153726976, 2401, 625000

Offset: 1

Author

Michel Marcus, Oct 21 2023

Keywords

• nonn

Comments

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

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000
• Jeffrey C. Lagarias and Wijit Yangjit, The factorial function and generalizations, extended, arXiv:2310.12949 [math.NT], 2023. See Section 7.2 pp. 20-21 and Table 1 p. 29.
• Wikipedia, p-adic valuation.

Crossrefs

Cf. A363838 (also uses gamma), A000027, A005451, A381885 (a(n)/n).

Programs

Maple

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

Mathematica

Table[Product[k^IntegerExponent[n, k], {k, 2, n}], {n, 1, 50}] (* Peter Luschny, Apr 01 2025 *)

PARI

f(n, b) = sum(i=1, logint(n, b), n\b^i);
d(n,b) = f(n,b)-f(n-1,b);
a(n) = prod(b=2, n, b^d(n,b));

Formula

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).

Extensions

More explicit name from Peter Luschny, Apr 01 2025