A085078 The largest number with the prime signature of n! using primes <= n.

1, 2, 6, 54, 750, 11250, 360150, 123531450, 3088286250, 64854011250, 77201350992150, 65389544290351050, 32637304517036749530, 2121424793607388719450, 163349709107768931397650

Offset: 1

Author

Amarnath Murthy, Jul 01 2003

Keywords

• nonn

Comments

n! is the smallest number with that prime signature. E.g. 720 = 2^4*3^2*5. (Can we name a(n) as the eldest brother of n!?) Subsidiary sequence: Total number of distinct numbers with prime signature that of n! having prime divisors less than or equal to n.

From Reikku Kulon, Sep 18 2008: (Start)

This is n! with prime exponents reversed. Perhaps it should be denoted with an inverted exclamation mark: (inverted-!)n

7! = 5040 = 2^4 * 3^2 * 5^1 * 7^1

(inverted-!)7 = 360150 = 2^1 * 3^1 * 5^2 * 7^4 (End)

Examples

			For n=6, 6!= 720 = 2^4*3^2*5, hence a(6) = 5^4*3^2*2 = 11250.
For n=8, 8! = 40320 = 2^7*3^2*5*7, hence a(8) = 7^7*5^2*3*2 = 123531450.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..181
• Index entries for sequences related to prime signature

Crossrefs

Cf. A000142, A069799.

Cf. A000040 [From Reikku Kulon, Sep 18 2008]

Programs

Mathematica

a[n_] := Module[{f = FactorInteger[n!], p, e}, p = First /@ f; e = Last /@ f; Times @@ (p^Reverse[e])]; Array[a, 15] (* Amiram Eldar, Dec 30 2020 *)

PARI

for (n = 1, 20, f = factor(n!); c = matsize(f)[1]; a = prod(i = 1, c, f[i, 1]^f[c + 1 - i, 2]); print(a)); \\ David Wasserman, Jan 14 2005

Formula

a(n) = A069799(A000142(n)). - Amiram Eldar, Dec 30 2020

Extensions

More terms from David Wasserman, Jan 14 2005