A102068 -id:A102068 - cp's OEIS frontend

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

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

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 *)

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

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

A122380 Numbers k such that k^2 > P(k)!, where P(k) is the greatest prime factor of k.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 243, 245

Offset: 1

Jonathan Sondow, Sep 03 2006

nonn

It is conjectured that k^2 < P(k)! for almost all k.

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

Charles R Greathouse IV, 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, A006530, A057109, A102068, A122378, A122379.

Reap[For[n = 2, n <= 250, n++, If[n^2 > FactorInteger[n][[-1, 1]]!, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2019 *)

smooth(P:vec, lim)=my(v=List([1]), nxt=vector(#P, i, 1), indx, t); while(1, t=vecmin(vector(#P, i, v[nxt[i]]*P[i]), &indx); if(t>lim, break); if(t>v[#v], listput(v, t)); nxt[indx]++); Vec(v)
list(lim)=my(v=List([2]),u,lower,upper=2,p=2); while(1, lower=upper+1; p=nextprime(p+1); upper=min(sqrtint(p!), lim); if(lower>lim, break); u=select(q->q>=lower, smooth(primes([2,p-1]),upper)); for(i=1,#u, listput(v,u[i]))); Vec(v) \\ Charles R Greathouse IV, Nov 09 2021

A333978 Numbers of the form b_1 * b_2 * ... * b_t, where b_1 = 1 and b_(i + 1) - b_i = 0 or 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 120, 128, 144, 162, 192, 216, 240, 256, 288, 324, 360, 384, 432, 480, 486, 512, 576, 600, 648, 720, 768, 864, 960, 972, 1024, 1080, 1152, 1200, 1296, 1440, 1458, 1536, 1728, 1800, 1920, 1944, 2048

Offset: 1

Peter Kagey, Sep 20 2020

This sequence gives the distinct values in A284001, sorted.
If m and k are in this sequence, then so is their product m*k.
If a prime p divides a(n), then so does p!.
A001013 is a subsequence.
Define a set S of polynomials by: (i) 1 is in S; (ii) if P is in S, then x*P and dP/dx are in S; (iii) if the repeated application of (i) and (ii) fails to prove that P is in S then P is not in S. This sequence enumerates the elements of S of degree 0. - Luc Rousseau, Aug 20 2022
Numbers k divisible by A102068(k)  (or in other words, numbers k divisible by h(k)! where h(k) is the largest prime factor of k). - David A. Corneth, Aug 20 2022

			The first 11 terms can be written as
   1 = 1
   2 = 1 * 2
   4 = 1 * 2 * 2
   6 = 1 * 2 * 3
   8 = 1 * 2 * 2 * 2
  12 = 1 * 2 * 2 * 3
  16 = 1 * 2 * 2 * 2 * 2
  18 = 1 * 2 * 3 * 3
  24 = 1 * 2 * 3 * 4 or 1 * 2 * 2 * 2 * 3
  32 = 1 * 2 * 2 * 2 * 2 * 2
  36 = 1 * 2 * 2 * 3 * 3

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Peter Kagey)

Cf. A001013, A003586, A006530, A102068, A284001, A334636.

is(n) = if(n==1, return(1)); my(f = factor(n), p = f[#f~, 1]); n%p! == 0 \\ David A. Corneth, Sep 05 2022

import heapq
from math import factorial
from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    oldv, h, primes, nextp, nextfact = 0, [(1, 1)], [], 0, 0
    while True:
        v, maxp = heapq.heappop(h)
        if v != oldv:
            yield v; oldv = v
            while nextfact < v:
                nextp = nextprime(nextp); nextfact = factorial(nextp)
                primes.append(nextp); heapq.heappush(h, (nextfact, nextp))
            for p in primes:
                if p <= maxp: heapq.heappush(h, (v*p, max(maxp, p)))
                else: break
print(list(islice(agen(), 60))) # Michael S. Branicky, Aug 20 2022

cp's OEIS Frontend

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

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

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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.

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A122380 Numbers k such that k^2 > P(k)!, where P(k) is the greatest prime factor of k.

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A333978 Numbers of the form b_1 * b_2 * ... * b_t, where b_1 = 1 and b_(i + 1) - b_i = 0 or 1.

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