OFFSET
1,1
COMMENTS
Numbers n > 1 for which there exists r <= gpf(n) such that r^k <= lpf(n) and gpf(n) < r^(k+1) for some k >= 0, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n) (the original, equivalent definition of the sequence).
Numbers n > 1 such that A252375(n) < 1 + A006530(n). Equally, one can substitute A251725 for A252375.
These are numbers n all of whose prime factors "fit between" two consecutive powers of some positive integer which itself is <= the largest prime factor of n.
Conjecture: If any n is in the sequence, then so is A003961(n).
Note: if Legendre's or Brocard's conjecture is true, then the above conjecture is true as well. See my comments at A251728. - Antti Karttunen, Jan 01 2015
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Ratio A251726(n)/A251727(n) (plotted with OEIS-server's Plot2-utility)
FORMULA
EXAMPLE
For 35 = 5*7, 7 is less than 5^2, thus 35 is included.
For 90 = 2*3*3*5, 5 is not less than 2^2, thus 90 is NOT included.
For 105 = 3*5*7, 7 is less than 3^2, thus 105 is included.
MATHEMATICA
pfQ[n_]:=Module[{f=FactorInteger[n]}, f[[-1, 1]]<f[[1, 1]]^2]; Select[ Range[ 200], pfQ] (* Harvey P. Dale, May 01 2015 *)
PROG
(Scheme with Antti Karttunen's IntSeq-library, three alternative versions)
(PARI) for(n=2, 150, if(vecmax(factor(n)[, 1]) < vecmin(factor(n)[, 1])^2, print1(n, ", "))) \\ Indranil Ghosh, Mar 24 2017
(Python)
from sympy import primefactors
print([n for n in range(2, 150) if max(primefactors(n))<min(primefactors(n))**2]) # Indranil Ghosh, Mar 24 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 17 2014
EXTENSIONS
A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section.
STATUS
approved