A057109 -id:A057109 - cp's OEIS frontend

A360169 Numbers k such that the partition number p(k) = A000041(k) is a term of A057109, i.e., it is not a divisor of the factorial of its greatest prime factor.

14, 19, 24, 28, 118

Offset: 1

Pontus von Brömssen, Jan 28 2023

There are no more terms below 3000.

			p(14) = 3^3 * 5, but 5! has only one factor 3.
p(19) = 2 * 5 * 7^2, but 7! has only one factor 7.
p(24) = 3^2 * 5^2 * 7, but 7! has only one factor 5.
p(28) = 2 * 11 * 13^2, but 13! has only one factor 13.
p(118) = 11 * 197 * 827^2, but 827! has only one factor 827.

Cf. A000041, A057109, A071963.

A002034 Kempner numbers: smallest positive integer m such that n divides m!.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

			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.

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.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 2.5.18 on page 77.

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.
Index entries for sequences related to factorial numbers.

Cf. A000142, A001113, A006530, A007672, A046022, A057109, A064759, A084945, A094371, A094372, A094404, A122378, A122379, A122416, A122417, A248937 (Fermi-Dirac analog: use unique representation of n > 1 as a product of distinct terms of A050376).

See A339594-A339596 for higher-dimensional generalizations.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a002034 1 = 1
a002034 n = fromJust (a092495 n `elemIndex` a000142_list)
-- Reinhard Zumkeller, Aug 24 2011

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

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 *)

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

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

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)Charles R Greathouse IV, Jul 30 2013

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

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

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

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

A070003 Numbers divisible by the square of their largest prime factor.

Original entry on oeis.org

4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 75, 81, 98, 100, 108, 121, 125, 128, 144, 147, 150, 162, 169, 196, 200, 216, 225, 242, 243, 245, 250, 256, 288, 289, 294, 300, 324, 338, 343, 361, 363, 375, 392, 400, 432, 441, 450, 484, 486, 490, 500, 507

Offset: 1

Labos Elemer, May 07 2002

nonn

Numbers n such that P(phi(n)) - phi(P(n)) = 1, where P(x) is the largest prime factor of x. P(phi(n)) - phi(P(n)) = A006530(A000010(n)) - A000010(A006530(n)).
Numbers n such that the value of the commutator of phi and P functions at n is -1.
Equivalently, n such that n and phi(n) have the same largest prime factor since Phi(p) = p-1 if p is prime. - Benoit Cloitre, Jun 08 2002
Since n is divisible by P(n)^2, n cannot divide P(n)! and so A057109 is a supersequence. Hence all A002034(a(n)) are composite. - Jonathan Sondow, Dec 28 2004
A225546 defines a self-inverse bijection between this sequence and A335740, considered as sets. - Peter Munn, Jul 19 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355. [alternate link]
Paul Erdős and Ilias Kastanas, Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
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."
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Totient Function

Subsequence of A057109, A122145.
Complement within A020725 of A102750.
Cf. A000010, A006530, A068211, A070777, A070812, A070002, A070004, A007283, A070813, A070814, A070815, A070816, A002034, A102067, A102068.
Related to A335740 via A225546.
A195212 is a subsequence.
Cf. A319988 (characteristic function). Positions of odd terms > 1 in A122111.

isA070003 := proc(n)
    if modp(n,A006530(n)^2) = 0 then # code re-use
        true;
    else
        false;
    end if;
end proc:
A070003 := proc(n)
    option remember ;
    if n =1 then
        4;
    else
        for a from procname(n-1)+1 do
            if isA070003(a) then
                return a
            end if;
        end do:
    end if;
end proc:
seq( A070003(n),n=1..80) ; # R. J. Mathar, Jun 27 2024

p[n_] := FactorInteger[n][[-1, 1]]; ep[n_] := EulerPhi[n]; fQ[n_] := p[ep[n]] == 1 + ep[p[n]]; Select[ Range[ 510], fQ] (* Robert G. Wilson v, Mar 26 2012 *)
Select[Range[500], FactorInteger[#][[-1,2]] > 1 &] (* T. D. Noe, Dec 06 2012 *)

for(n=3,1000,if(component(component(factor(n),1),omega(n))==component(component(factor(eulerphi(n)),1),omega(eulerphi(n))),print1(n,",")))

is(n)=my(f=factor(n)[,2]);f[#f]>1 \\ Charles R Greathouse IV, Mar 21 2012

sm(lim,mx)=if(mx==2,return(vector(log(lim+.5)\log(2)+1,i,1<<(i-1))));my(v=[1]);forprime(p=2,min(mx,lim),v=concat(v,p*sm(lim\p,p)));vecsort(v)
list(lim)=my(v=[]);forprime(p=2,sqrt(lim),v=concat(v,p^2*sm(lim\p^2,p)));vecsort(v) \\ Charles R Greathouse IV, Mar 27 2012

from sympy import factorint
def ok(n): f = factorint(n); return f[max(f)] >= 2
print(list(filter(ok, range(4, 508)))) # Michael S. Branicky, Apr 08 2021

Erdős proved that there are x * exp(-(1 + o(1))sqrt(log x log log x)) members of this sequence up to x. - Charles R Greathouse IV, Mar 26 2012

New name from Jonathan Sondow and Charles R Greathouse IV, Mar 27 2012

A102068 a(n) = P(n)!, where P(n) is the largest prime factor of n (with a(1) = 1).

Original entry on oeis.org

1, 2, 6, 2, 120, 6, 5040, 2, 6, 120, 39916800, 6, 6227020800, 5040, 120, 2, 355687428096000, 6, 121645100408832000, 120, 5040, 39916800, 25852016738884976640000, 6, 120, 6227020800, 6, 5040, 8841761993739701954543616000000, 120

Offset: 1

Jonathan Sondow, Dec 28 2004

nonn

P(n)! is a multiple of n, for almost all n. The exceptions are A057109.

			P(12)! = 3! = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..456
Paul Erdős and Ilias Kastanas, Solution 6674: The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
A. Ivic, On a problem of Erdos involving the largest prime factor of n, arXiv:math/0311056 [math.NT], 2003-2004.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for sequences related to factorial numbers.

Cf. A000040, A000142, A006530, A057109, A061395, A102067.

Table[FactorInteger[n][[-1,1]]!,{n,30}] (* Harvey P. Dale, Jan 29 2014 *)

a(n) = if (n==1, 1, vecmax(factor(n)[,1])!); \\ Michel Marcus, Sep 24 2022

a(n) = A000142(A006530(n)) = A000040(A061395(n))!.

A122378 Numbers m such that m^2 > S(m)!, where S(m)! is the smallest factorial divisible by m.

Original entry on oeis.org

2, 3, 6, 8, 12, 15, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 224, 240, 252, 280, 288, 315, 320, 336, 360, 384, 420, 448, 480, 504, 560, 576, 630, 640, 648, 672, 720, 756, 810, 840, 864, 896, 945, 960, 1008, 1080

Offset: 1

Jonathan Sondow, Sep 03 2006

nonn

It is conjectured that m^2 < S(m)! for almost all m.

For each k > 1, at most tau(k!)/2 = A000005(k!)/2 are in the sequence because of that k. So at most Sum_{k = 1..m} tau(k!)/(2*m!) of the numbers up to m! are terms. This tends to 0 as m tends to infinity. - David A. Corneth, Dec 29 2019

			15^2 = 225 > 120 = 5! = S(15)!, so 15 is a member.

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function
Index entries for sequences related to factorial numbers.

Cf. A000290, A002034, A057109, A092495, A122379, A122380.

nmax = 1100;
Do[m = 1; While[!IntegerQ[m!/n], m++]; S[n] = m, {n, 1, nmax}];
Select[Range[nmax], #^2 > S[#]!&] (* Jean-François Alcover, Dec 04 2018 *)

upto(n) = {my(res = List(), maxf = 1, olddiv, newdiv, n2 = n^2, cf = 1); while(maxf! < n2, maxf++); maxf--; olddiv = divisors(0!); newdiv = divisors(1!); for(i = 2, maxf, olddiv = newdiv; cf*=i; newdiv = divisors(cf); cans = setminus(Set(newdiv), Set(olddiv)); for(j = 1, #cans, if(cans[j]^2 > cf, if(cans[j] <= n, listput(res, cans[j]) , next(2) ); ) ) ); listsort(res); res } \\ David A. Corneth, Dec 29 2019

A122379 Numbers n such that S(n)! > n^2 > P(n)!, where S(n)! is the smallest factorial divisible by n and P(n) is the greatest prime factor of n.

Original entry on oeis.org

4, 9, 16, 18, 25, 27, 32, 50, 54, 64, 75, 81, 96, 98, 100, 108, 125, 128, 135, 147, 150, 160, 162, 175, 189, 192, 196, 200, 216, 225, 243, 245, 250, 256, 270, 294, 300, 324, 343, 350, 375, 378, 392, 400, 405, 432, 441, 450, 486, 490, 500, 512, 525, 540, 567

Offset: 1

Jonathan Sondow, Sep 03 2006

nonn

It is conjectured that n^2 < P(n)! for almost all n. It is known that S(n) = P(n) for almost all n. Clearly, S(n) >= P(n) for all n > 1.

			S(9)! = 6! = 720 > 81 = 9^2 > 6 = 3! = P(9)!, so 9 is a member.

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
Index entries for sequences related to factorial numbers.

Cf. A000290, A002034, A006530, A057109, A092495, A102068, A122378, A122380.

s[n_] := For[k = 1, True, k++, If[Divisible[k!, n], Return[k]]];
p[n_] := FactorInteger[n][[-1, 1]];
okQ[n_] := s[n]! > n^2 > p[n]!;
Select[Range[1000], okQ] (* Jean-François Alcover, Jan 27 2019 *)

A362333 Least nonnegative integer k such that (gpf(n)!)^k is divisible by n, where gpf(n) is the greatest prime factor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Pontus von Brömssen, Apr 16 2023

nonn

First differs from A088388 at n = 40.

			For n = 12, gpf(n)! = 3! = 6 is not divisible by 12, but (3!)^2 = 36 is divisible by 12, so a(12) = 2.

Cf. A006530, A051903, A057109, A088388, A371148, A371149, A371151, A371152.

from sympy import factorint
def A362333(n):
    f = factorint(n)
    gpf = max(f,default=None)
    a = 0
    for p in f:
        m = gpf
        v = 0
        while m >= p:
            m //= p
            v += m
        a = max(a,-(-f[p]//v))
    return a

a(n) > 1 if and only if n is in A057109.
a(n) <= A051903(n).
a(n) = ceiling(A371148(n)/A371149(n)). - Pontus von Brömssen, Mar 16 2024

A048839 Numbers k dividing P(k)!, where P(k) is the largest prime factor of k.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

Solutions to Diophantine equation P(k) = K(k), where P(k) is the largest prime which divides k and K(k) is the Kempner function A002034.

P. Erdős and C. Ashbacher, Thoughts of Pal Erdos on Some Smarandache Notions, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 220-224.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement is A057109.

Cf. A002034, A102068.

Select[Range[2, 100], Divisible[FactorInteger[#][[-1, 1]]!, #] &] (* Amiram Eldar, May 23 2024 *)

is(n)=my(s=factor(n)[, 1]); s[#s]!%n==0 \\ Charles R Greathouse IV, Sep 20 2012

a(n) ~ n. - Charles R Greathouse IV, Sep 20 2012

Corrected and extended by Naohiro Nomoto, Jul 14 2001

A057108 Difference between the smallest number S(n) with S(n)! a multiple of n and the largest prime factor of n [taking a(1)=0].

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 1, 0, 0, 0, 4, 0, 3, 0, 0, 0, 0, 0, 1, 5, 0, 6, 0, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 7, 5, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 5, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 7, 0, 5, 0, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Aug 08 2000

			a(12)=1 since 12 is a factor of 4!, 3 is the largest prime factor of 12 and 4-3=1

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 284-292.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Steven R. Finch, The Average Value of the Smarandache Function [Broken link]
Steven R. Finch, The Average Value of the Smarandache Function

Cf. A002034, A006530, A057109, A057110.

A002034(n) = if(1==n,n, my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); (k)); \\ After code in A002034.
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
A057108(n) = (A002034(n) - A006530(n)); \\ Antti Karttunen, Jul 22 2018

a(n) = A002034(n) - A006530(n).

More terms from James Sellers, Aug 22 2000

A102067 Numbers k such that k does not divide P(k)! even though P(k)^2 is not a factor of k, where P(k) is the largest prime factor of k.

Original entry on oeis.org

12, 24, 45, 48, 80, 90, 96, 135, 160, 175, 180, 189, 192, 224, 240, 270, 320, 350, 360, 378, 384, 405, 448, 480, 525, 539, 540, 567, 637, 640, 672, 700, 720, 756, 768, 810, 875, 896, 945, 960, 1050, 1078, 1080, 1120, 1134, 1215, 1274, 1280, 1344, 1375, 1400, 1440

Offset: 1

Jonathan Sondow, Dec 28 2004

nonn

Clearly, if P(k)^2 is a factor of k, then k does not divide P(k)!. Each member shows that the converse is false.

k is a member if and only if k is in A057109 but not in A070003.

			12 does not divide P(12)! = 3! and 3^2 is not a factor of 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
Aubrey J. Kempner, Miscellanea, Amer. Math. Monthly, Vol. 25, No. 5 (1918), pp. 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Index entries for sequences related to factorial numbers.

Cf. A006530, A057109, A070003, A102068.

q[n_] := Module[{p = FactorInteger[n][[-1, 1]]}, !Divisible[n, p^2] && !Divisible[p!, n]]; Select[Range[1500], q] (* Amiram Eldar, Mar 30 2021 *)

isok(n) = {my(f = factor(n)); my(P = f[#f~,1]); (P! % n) && (n % P^2);} \\ Michel Marcus, Sep 16 2015

More terms from Michel Marcus, Sep 16 2015

cp's OEIS Frontend

A360169 Numbers k such that the partition number p(k) = A000041(k) is a term of A057109, i.e., it is not a divisor of the factorial of its greatest prime factor.

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A002034 Kempner numbers: smallest positive integer m such that n divides m!.

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A070003 Numbers divisible by the square of their largest prime factor.

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A102068 a(n) = P(n)!, where P(n) is the largest prime factor of n (with a(1) = 1).

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A122378 Numbers m such that m^2 > S(m)!, where S(m)! is the smallest factorial divisible by m.

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A122379 Numbers n such that S(n)! > n^2 > P(n)!, where S(n)! is the smallest factorial divisible by n and P(n) is the greatest prime factor of n.

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A362333 Least nonnegative integer k such that (gpf(n)!)^k is divisible by n, where gpf(n) is the greatest prime factor of n.

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