A251726 Numbers n > 1 for which gpf(n) < lpf(n)^2, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n).

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 96, 97, 101, 103, 105, 107, 108, 109, 113, 115, 119, 121, 125, 127, 128, 131, 133, 135, 137, 139, 143, 144

Offset: 1

Author

Antti Karttunen, Dec 17 2014

Keywords

• nonn

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

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Ratio A251726(n)/A251727(n) (plotted with OEIS-server's Plot2-utility)

Crossrefs

Complement: A251727. Subsequences: A251728, A000961 (after 1).

Characteristic function: A252372. Inverse function: A252373.

Gives the positions of zeros in A252459 (after its initial zero), cf. also A284261.

Cf. A252370 (gives the difference between the prime indices of gpf and lpf for each a(n)).

Sequence gives all n > 1 for which A284252(n) (equally: A284254) is 1, and A284256(n) (equally A284258) is 0, and also n > 1 such that A284260(n) = A006530(n).

Cf. A003961, A006530, A020639, A252375, A251725, A253784.

Related permutations: A252757-A252758.

Programs

Mathematica

pfQ[n_]:=Module[{f=FactorInteger[n]},f[[-1,1]]Harvey P. Dale, May 01 2015 *)

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))Indranil Ghosh, Mar 24 2017

Formula

Other identities. For all n >= 1:

A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]

Extensions

A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section.