login
Least inverse of the Kempner function A002034.
9

%I #38 Oct 17 2024 14:28:47

%S 1,2,3,4,5,9,7,32,27,25,11,243,13,49,125,4096,17,2187,19,625,343,121,

%T 23,59049,3125,169,177147,2401,29,78125,31,134217728,1331,289,16807,

%U 43046721,37,361,2197,1953125,41,117649,43,14641,9765625,529,47

%N Least inverse of the Kempner function A002034.

%C 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

%C 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

%D 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.

%H Charlie Neder, <a href="/A046021/b046021.txt">Table of n, a(n) for n = 1..1000</a>

%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorial.html">Factorial</a>.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F 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

%t 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 *)

%o (PARI) A046021(n,p=A006530(n))=p^A102048(n,p) \\ _M. F. Hasler_, Nov 27 2018

%o (Python)

%o from sympy import primefactors, integer_log

%o def A046021(n):

%o if n == 1: return 1

%o p = max(primefactors(n))

%o 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

%Y Cf. A002034, A046022, A006530, A102048.

%K nonn,nice

%O 1,2

%A _Eric W. Weisstein_

%E More terms from _David W. Wilson_ and _Christian G. Bower_, independently.