A046021 Least inverse of the Kempner function A002034.

1, 2, 3, 4, 5, 9, 7, 32, 27, 25, 11, 243, 13, 49, 125, 4096, 17, 2187, 19, 625, 343, 121, 23, 59049, 3125, 169, 177147, 2401, 29, 78125, 31, 134217728, 1331, 289, 16807, 43046721, 37, 361, 2197, 1953125, 41, 117649, 43, 14641, 9765625, 529, 47

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice

Comments

To compute the n-th term for n > 1: For each prime p that divides n, find the highest power p^E(p) that divides (n-1)!. Then a(n) is the smallest of the numbers p^(E(p)+1). - Jonathan Sondow, Mar 03 2004

p^(E(p)+1) is smallest when p = P(n), the largest prime dividing n (since E(p) is approximately p^((n-1)/(p-1)), which decreases as p increases). So a(n) = P(n)^(E(P(n))+1) = A006530(n)^A102048(n) for n>1. - Jonathan Sondow, Dec 26 2004

References

• R. L. Graham, D. E. Knuth and O. Patashnik, "Factorial Factors" Sect. 4.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 111-115, 1994.

Links

• Charlie Neder, Table of n, a(n) for n = 1..1000
• J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function.
• Eric Weisstein's World of Mathematics, Greatest Prime Factor.
• Eric Weisstein's World of Mathematics, Factorial.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A002034, A046022, A006530, A102048.

Programs

Mathematica

With[{p=First[Last[FactorInteger[n, FactorComplete->True]]]}, p^(1+Sum[Floor[(n-1)/p^k], {k, Floor[Log[n-1]/Log[p]]}])] (* Jonathan Sondow, Dec 26 2004 *)

PARI

A046021(n,p=A006530(n))=p^A102048(n,p) \\ M. F. Hasler, Nov 27 2018

Python

from sympy import primefactors, integer_log
def A046021(n):
    if n == 1: return 1
    p = max(primefactors(n))
    return p**sum(((n-1)//p**k for k in range(1,integer_log(n-1,p)[0]+1)),start=1) # Chai Wah Wu, Oct 17 2024

Formula

a(n) = P^(1+Sum(k=1 to [log(n-1)/log(P)], [(n-1)/P^k])) for n>1, where P = A006530(n) is the largest prime dividing n. E.g. a(6) = 9 because 9 is the least integer m with A002034(m) = 6, that is, m divides 6!, but m does not divide k! for k < 6. - Jonathan Sondow, Dec 26 2004

Extensions

More terms from David W. Wilson and Christian G. Bower, independently.