cp's OEIS Frontend

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

%I A122380 #24 Aug 10 2025 20:53:32
%S A122380 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,
%T A122380 75,80,81,84,90,96,98,100,105,108,112,120,125,126,128,135,140,144,147,
%U A122380 150,160,162,168,175,180,189,192,196,200,210,216,224,225,240,243,245
%N A122380 Numbers k such that k^2 > P(k)!, where P(k) is the greatest prime factor of k.
%C A122380 It is conjectured that k^2 < P(k)! for almost all k.
%H A122380 Charles R Greathouse IV, <a href="/A122380/b122380.txt">Table of n, a(n) for n = 1..10000</a>
%H A122380 J. Sondow, <a href="http://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.
%H A122380 J. Sondow, <a href="http://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.
%H A122380 J. Sondow and E. W. Weisstein, <a href="https://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</a>
%H A122380 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers.</a>
%e A122380 15^2 = 225 > 120 = 5! = P(15)!, so 15 is a member.
%t A122380 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 *)
%o A122380 (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)
%o A122380 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
%Y A122380 Cf. A000290, A006530, A057109, A102068, A122378, A122379.
%K A122380 nonn
%O A122380 1,1
%A A122380 _Jonathan Sondow_, Sep 03 2006