

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).


17



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
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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 OEISserver's Plot2utility)


FORMULA

Other identities. For all n >= 1:
A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]


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 IntSeqlibrary, three alternative versions)
(define A251726 (MATCHINGPOS 1 2 (lambda (n) (< (A006530 n) (A000290 (A020639 n))))))
(define A251726 (MATCHINGPOS 1 2 (lambda (n) (< (A251725 n) (+ 1 (A006530 n))))))
(define A251726 (MATCHINGPOS 1 2 (lambda (n) (< (A252375 n) (+ 1 (A006530 n))))))
(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 xrange(2, 150) if max(primefactors(n))<min(primefactors(n))**2] # Indranil Ghosh, Mar 24 2017


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: A252757A252758.
Sequence in context: A253567 A048197 A253784 * A193671 A073491 A066311
Adjacent sequences: A251723 A251724 A251725 * A251727 A251728 A251729


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



