A251726 - OEIS

%I #54 Mar 13 2021 00:56:00

%S 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,

%T 36,37,41,43,45,47,48,49,53,54,55,59,61,63,64,65,67,71,72,73,75,77,79,

%U 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

%N 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).

%C 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).

%C Numbers n > 1 such that A252375(n) < 1 + A006530(n). Equally, one can substitute A251725 for A252375.

%C 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.

%C Conjecture: If any n is in the sequence, then so is A003961(n).

%C 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

%H Antti Karttunen, <a href="/A251726/b251726.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="https://oeis.org/plot2a?name1=A251726&amp;name2=A251727&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Ratio A251726(n)/A251727(n)</a> (plotted with OEIS-server's Plot2-utility)

%F Other identities. For all n >= 1:

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

%e For 35 = 5*7, 7 is less than 5^2, thus 35 is included.

%e For 90 = 2*3*3*5, 5 is not less than 2^2, thus 90 is NOT included.

%e For 105 = 3*5*7, 7 is less than 3^2, thus 105 is included.

%t pfQ[n_]:=Module[{f=FactorInteger[n]},f[[-1,1]]<f[[1,1]]^2]; Select[ Range[ 200],pfQ] (* _Harvey P. Dale_, May 01 2015 *)

%o (Scheme with _Antti Karttunen_'s IntSeq-library, three alternative versions)

%o (define A251726 (MATCHING-POS 1 2 (lambda (n) (< (A006530 n) (A000290 (A020639 n))))))

%o (define A251726 (MATCHING-POS 1 2 (lambda (n) (< (A251725 n) (+ 1 (A006530 n))))))

%o (define A251726 (MATCHING-POS 1 2 (lambda (n) (< (A252375 n) (+ 1 (A006530 n))))))

%o (PARI) for(n=2, 150, if(vecmax(factor(n)[,1]) < vecmin(factor(n)[,1])^2, print1(n,", "))) \\ _Indranil Ghosh_, Mar 24 2017

%o (Python)

%o from sympy import primefactors

%o print([n for n in range(2, 150) if max(primefactors(n))<min(primefactors(n))**2]) # _Indranil Ghosh_, Mar 24 2017

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

%Y Characteristic function: A252372. Inverse function: A252373.

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

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

%Y 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).

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

%Y Related permutations: A252757-A252758.

%K nonn

%O 1,1

%A _Antti Karttunen_, Dec 17 2014

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