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

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

1,1

Comments

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

Links

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.

Example

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A000290, A006530, A057109, A102068, A122378, A122379.

Sequence in context: A376154 A145807 A278962 * A033501 A336504 A331827

Adjacent sequences: A122377 A122378 A122379 * A122381 A122382 A122383

Keyword

nonn

Author

Jonathan Sondow, Sep 03 2006

Status

approved