OFFSET
1,2
COMMENTS
Sometimes named after Florentin Smarandache, although studied 60 years earlier by Aubrey Kempner and 35 years before that by Lucas.
Kempner originally defined a(1) to be 0, and there are good reasons to prefer that (see Hungerbühler and Specker), but we shall stay with the by-now traditional value a(1) = 1. - N. J. A. Sloane, Jan 02 2021
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) is the 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
REFERENCES
E. Lucas, Question Nr. 288, Mathesis 3 (1883), 232.
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 proposées, Question Nr. 288, Mathesis 7 (1887), 68-69.
Donald J. Newman, A Problem Seminar. Problem 17, Springer-Verlag, 1982.
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).
Florentin Smarandache, A Function in the Number Theory, Analele Univ. Timisoara, Fascicle 1, Vol. XVIII, 1980, pp. 79-88; Smarandache Function J., Vol. 1, No. 1-3 (1990), pp. 3-17.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Charles Ashbacher, An Introduction to the Smarandache Function, Erhus Univ. Press, Vail, 62 pages, 1995 [WaybackMachine cached version added by Felix Fröhlich, Sep 10 2018].
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 147.
Constantin Dumitrescu, A brief history of the "Smarandache function", Bull. Pure Appl. Sci. Sec. E, Math., Vol. 12, No. 1-2 (1993), pp. 77-82 [WaybackMachine cached version added by Felix Fröhlich, Sep 10 2018].
Constantin Dumitrescu and Vasile Seleacu, The Smarandache Function, Erhus Univ. Press, Vail, 137 pages, 1996 [WaybackMachine cached version added by Felix Fröhlich, Sep 10 2018].
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), p. 246.
Paul Erdős and Ilias Kastanas, Solution 6674: The smallest factorial that is a multiple of n, Amer. Math. Monthly, Vol. 101, No. 2 (1994) p. 179.
Steven R. Finch, The average value of the Smarandache function, Smarandache Notions Journal, Vol. 10, No. 1-3 (1999), pp. 95-96.
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Norbert Hungerbühler and Ernst Specker, A generalization of the Smarandache Function to several variables, INTEGERS, Vol. 6 (2006), #A23
Aleksandar Ivić, On a problem of Erdős involving the largest prime factor of n, arXiv:math/0311056 [math.NT], 2003-2004.
Aubrey J. Kempner, Miscellanea, The American Mathematical Monthly, Vol. 25, No. 5 (May, 1918), pp. 201-210. (See Section II, Concerning the smallest integer m! divisible by a given integer n.)
G. Kreweras, Sur quelques problèmes relatifs au vote pondéré, [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), pp. 45-63.
Xiumei Li and Min Sha, A proof of Sondow's conjecture on the Smarandache function, arXiv preprint arXiv:1907.00370 [math.NT], 2019-2020. Amer. Math. Monthly, 127:10 (2020), 939-943.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., Vol. 35 (1979), pp. 1-16. [From N. J. A. Sloane, Feb 07 2009]
Jon Perry, Calculating the Smarandache Numbers, Smarandache Notions Journal, Vol. 14, No. 1 (2004), pp. 124-127.
Project Euler, Problem 549: Divisibility of factorials.
Sebastián Martín Ruiz, A Congruence with Smarandache's Function, Smarandache Notions Journal, Vol. 10, No. 1-3 (1999), pp. 130-132.
József Sándor, The Smarandache minimum and maximum functions, Scientia Magna, Vol. 1, No. 2 (2005), pp. 162-166. See p. 164.
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, Vol. 113 (2006), pp. 637-641 and Vol. 114 (2007), p. 659.
Jonathan Sondow and Kyle 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.
Jonathan Sondow and Eric W. Weisstein, MathWorld: Smarandache Function.
J. Thompson, A propriety (sic) of smarandache function, Abstract 878-11-758, Notices Amer. Math. Soc., Vol. 14 (1993), p. 41. [Annotated scanned copy]
Wikipedia, Kempner function.
FORMULA
From Joerg Arndt, Jul 14 2012: (Start)
The following identities were 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 (Erdős 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
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) is the 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
Conjecture: a(n) = 1 + n - Sum_{k=1..n} Sum_{m=1..n} cos(-2*Pi*k/n*m!)/n. Formula verified for the first 500 terms. - Mats Granvik, Feb 26 2021
Limit_{n->oo} (1/n) * Sum_{k=2..n} log(a(k))/log(k) = A084945 (Finch, 1999). - Amiram Eldar, Jul 04 2021
EXAMPLE
1! = 1, but clearly 8 does not divide 1.
2! = 2, but 8 does not divide 2.
3! = 6, but 8 does not divide 6.
4! = 24, and 8 does divide 24. Hence a(8) = 4.
However, 9 does not divide 24.
5! = 120, but 9 does not divide 120.
6! = 720, and 9 does divide 720. Hence a(9) = 6.
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
g:= proc(p, u)
local i, t;
t:= 0;
for i from 1 while t < u do
t:= t + 1 + padic[ordp](i, p);
od;
p*(i-1)
end;
A002034:= x -> max(map(g@op, ifactors(x)[2])); # Robert Israel, Apr 20 2014
MATHEMATICA
Do[m = 1; While[ !IntegerQ[m!/n], m++ ]; Print[m], {n, 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}] (* Eric W. Weisstein, based on a formula of Kempner's, May 17 2004 *)
With[{facts = Range[100]!}, Flatten[Table[Position[facts, _?(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
(Python)
from sympy import factorial
def a(n):
m=1
while True:
if factorial(m)%n==0: return m
else: m+=1
[a(n) for n in range(1, 101)] # Indranil Ghosh, Apr 24 2017
(Python)
from sympy import factorint
def valp(n, p):
s = 0
while n: n //= p; s += n
return s
def K(p, e):
if e <= p: return e*p
t = e*(p-1)//p*p
while valp(t, p) < e: t += p
return t
def A002034(n):
return 1 if n == 1 else max(K(p, e) for p, e in factorint(n).items())
print([A002034(n) for n in range(1, 85)]) # Michael S. Branicky, Jun 09 2022 after Charles R Greathouse IV
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Error in 45th term corrected by David W. Wilson, May 15 1997
STATUS
approved