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A002034 Kempner numbers: smallest number m such that n divides m!.
(Formerly M0453 N0167)
92
1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, 4, 13, 7, 5, 6, 17, 6, 19, 5, 7, 11, 23, 4, 10, 13, 9, 7, 29, 5, 31, 8, 11, 17, 7, 6, 37, 19, 13, 5, 41, 7, 43, 11, 6, 23, 47, 6, 14, 10, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 7, 8, 13, 11, 67, 17, 23, 7, 71, 6, 73, 37, 10, 19, 11, 13, 79, 6, 9, 41, 83, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Commonly named after Florentin Smarandache, although studied 60 years earlier by Aubrey Kempner and 35 years before that by Lucas.

Kempner gave an algorithm to compute a(n) from the prime factorization of n. Partial solutions were given earlier by Lucas in 1883 and Neuberg in 1887. - Jonathan Sondow, Dec 23 2004

a(n) = degree of lowest degree monic polynomial over Z that vanishes identically on the integers mod n [Newman]

Smallest k such that n divides product of k consecutive integers starting with n+1. - Amarnath Murthy, Oct 26 2002

If m and n are any integers with n > 1, then |e - m/n| > 1/(a(n)+1)! (see Sondow 2006).

Degree of minimal linear recurrence satisfied by Bell numbers (A000110) read modulo n. [Lunnon et al.] - N. J. A. Sloane, Feb 07 2009

A000142(a(n)) = A092495(n). [Reinhard Zumkeller, Aug 24 2011]

REFERENCES

C. Dumitrescu, A brief history of the "Smarandache function". Bull. Pure Appl. Sci. Sec. E, Math., 12 (1993), no. 1-2, 77-82.

P. Erdos and I. Kastanas, Problem/Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.

E. Hungerbuhler, E. Specker, A generalization of the Smarandache Function to several variables, INTEGERS 6 (2006) #A23

A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n".

E. Lucas, Question Nr. 288, Mathesis 3 (1883), 232.

W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16. [From N. J. A. Sloane, Feb 07 2009]

R. Muller, Unsolved problems related to Smarandache Function, Number Theory Publishing Company, Phoenix, AZ, ISBN 1-879585-37-5, 1993.

J. Neuberg, Solutions des questions proposees, Question Nr. 288, Mathesis 7 (1887), 68-69.

D. J. Newman, A Problem Seminar. Problem 17, Springer-Verlag 1982.

S. M. Ruiz, A Congruence with Smarandache's Function, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 130-132.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. Smarandache, A Function in the Number Theory, Analele Univ. Timisoara, Fascicle 1, Vol. XVIII, 1980, pp. 79-88; Smarandache Function J. 1 (1990), no. 1, 3-17.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

C. Ashbacher, An Introduction to the Smarandache Function, Erhus Univ. Press, Vail, 62 pages, 1995.

C. Dumitrescu and V. Seleacu, The Smarandache Function, Erhus Univ. Press, Vail, 137 pages, 1996.

A. Ivic (2004), On a problem of Erdos involving the largest prime factor of n

J. Perry, Calculating the Smarandache Numbers [Broken link]

A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, Concerning the smallest integer m! divisible by a given integer n.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641 and 114 (2007), 659.

J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function

Index entries for sequences related to factorial numbers.

FORMULA

From Joerg Arndt, Jul 14 2012: (Start)

The following identities where given by Kempner (1918):

a(1) = 1.

a(n!) = n.

a(p) = p  for p prime.

a(p1*p2*...*pu) = pu  if p1 < p2 < ... < pu are distinct primes.

a(p^k) = p*k for p prime and k<=p.

Let n = p1^e1 * p2^e2 * ... * pu^eu be the canonical factorization of n, then a(n) = max( a(p1^e1),  a(p2^e2), ..., a(pu^eu) ).

(End)

Clearly a(n) >= P(n), the largest prime factor of n (= A006530). a(n) = P(n) for almost all n (Erdos and Kastanas 1994, Ivic 2004). The exceptions are A057109. a(n) = P(n) if and only if a(n) is prime because if a(n) > P(n) and a(n) were prime, then since n divides a(n)!, n would also divide (a(n)-1)!, contradicting minimality of a(n). - Jonathan Sondow, Jan 10 2005

It appears that if p is prime then a(p^k)=k*p for 0<=k<=p. Hence it appears also that if n = 2^m*p(1)^e(1)*...*p(r)^e(r) and if there exists b, 1<=b<= r, such that Max( 2*m+2, p(i)*e(i), 1<=i<=r ) = p(b)*e(b) with e(b)<=p(b) then a(n)=e(b)*p(b). E.g.: a(2145986896455317997802121296896)=a(2^10*3^3*7^9*11^9*13^8) = 13*8 = 104, since 8*13 = Max (2*10+2, 3*3, 7*9, 11*9, 13*8) and 8<=13. - Benoit Cloitre, Sep 01 2002

It appears that a(2^m-1) = largest prime factor of 2^m - 1 (A005420).

a(n!) = n for all n > 0 and a(p) = p if p is prime. - Jonathan Sondow, Dec 23 2004

EXAMPLE

a(8) = 4 because 8 divides 4! and 8 does not divide k! for k < 4.

MAPLE

a:=proc(n) local b: b:=proc(m) if type(m!/n, integer) then m else fi end: [seq(b(m), m=1..100)][1]: end: seq(a(n), n=1..84); # Emeric Deutsch, Aug 01 2005

MATHEMATICA

Do[m = 1; While[ !IntegerQ[m!/n], m++ ]; Print[m], {n, 1, 85}] (* or for larger n's *)

Kempner[1] := 1; Kempner[n_] := Max[Kempner @@@ FactorInteger[n]]; Kempner[p_, 1] := p; Kempner[p_, alpha_] := Kempner[p, alpha] = Module[{a, k, r, i, nu, k0 = alpha(p - 1)}, i = nu = Floor[Log[p, 1 + k0]]; a[1] = 1; a[n_] := (p^n - 1)/(p - 1); k[nu] = Quotient[alpha, a[nu]]; r[nu] = alpha - k[nu]a[nu]; While[r[i] > 0, k[i - 1] = Quotient[r[i], a[i - 1]]; r[i - 1] = r[i] - k[i - 1]a[i - 1]; i-- ]; k0 + Plus @@ k /@ Range[i, nu]]; Table[ Kempner[n], {n, 85}] (* from Eric Weisstein, based on a formula of Kempner's, May 17 2004 *)

With[{f=Range[100]!}, Flatten[Table[Position[f, _?(Divisible[#, n]&), {1}, 1], {n, 90}]]] (* Harvey P. Dale, May 24 2013*)

PROG

(PARI) a(n)=if(n<0, 0, s=1; while(s!%n>0, s++); s)

(PARI) a(n)=my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); k \\ Charles R Greathouse IV, Feb 28 2012

(PARI) valp(n, p)=my(s); while(n\=p, s+=n); s

K(p, e)=if(e<=p, return(e*p)); my(t=e*(p-1)\p*p); while(valp(t+=p, p)<e, ); t

a(n)=my(f=factor(n), m=1); for(i=1, #f~, m=max(K(f[i, 1], f[i, 2]), m)); m \\ Charles R Greathouse IV, Jul 30 2013

(Haskell)

import Data.List (elemIndex)

import Data.Maybe (fromJust)

a002034 1 = 1

a002034 n = fromJust (a092495 n `elemIndex` a000142_list)

-- Reinhard Zumkeller, Aug 24 2011

CROSSREFS

Cf. A007672, A064759, A000142, A094371, A094372, A046022, A094404, A006530, A057109, A001113, A122378, A122379, A122416, A122417.

Sequence in context: A025492 A077004 A064760 * A088491 A140271 A223491

Adjacent sequences:  A002031 A002032 A002033 * A002035 A002036 A002037

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Error in 45th term corrected by David W. Wilson May 15 1997

STATUS

approved

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Last modified April 16 14:51 EDT 2014. Contains 240600 sequences.