A093896 Least positive k such that n^n divides k!.

1, 4, 9, 10, 25, 15, 49, 28, 39, 45, 121, 28, 169, 91, 65, 66, 289, 78, 361, 85, 133, 231, 529, 76, 205, 325, 165, 175, 841, 125, 961, 164, 341, 561, 217, 150, 1369, 703, 481, 165, 1681, 259, 1849, 451, 186, 1035, 2209, 196, 595, 405, 833, 637, 2809, 327, 561

Offset

1,2

Comments

a(p) = p^2, p is a prime. a(n) <= n^2.

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Formula

a(n) = A002034(n^n). - Chai Wah Wu, Feb 26 2026

Mathematica

Do[k = 1; While[Mod[k!, n^n] > 0, k++ ]; Print[k], {n, 1, 30}] (* Ryan Propper, Jun 20 2005 *)

Prog

(Python)
from itertools import count
from sympy import factorint, multiplicity
def A093896(n):
    m = 1
    for p, e in factorint(n).items():
        g = e*n
        if g<=p:
            m = max(m, g*p)
        else:
            c, a = 0, max(p, (g*(p-1)//p**2)*p)
            b = a-p
            while b>0:
                c += (b:=b//p)
            for k in count(a, p):
                c += multiplicity(p, k) if p>2 else (~k&k-1).bit_length()
                if c+k+p > g:
                    m = max(m, p*(max(k, g-c) if g>=p else g+1-c))
                    break
    return m # Chai Wah Wu, Feb 26 2026

Crossrefs

Cf. A093897, A002034.

Sequence in context: A030754 A244863 A077584 * A332449 A191905 A113432

Adjacent sequences: A093893 A093894 A093895 * A093897 A093898 A093899

Keyword

nonn

Author

Amarnath Murthy, Apr 23 2004

Extensions

22 more terms from Ryan Propper, Jun 20 2005

More terms from Stefan Steinerberger, Feb 12 2006

More terms from Jon E. Schoenfield, Jan 06 2009

Status

approved