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A140269 Numbers n such that GCD((largest divisor of n that is <= sqrt(n)),(smallest divisor of n that is >=sqrt(n))) > 1. 2

%I

%S 4,8,9,16,18,24,25,27,32,36,48,49,50,54,60,64,75,80,81,96,98,100,108,

%T 112,120,121,125,128,135,140,144,147,150,160,162,168,169,180,189,192,

%U 196,200,216,224,225,242,243,245,250,252,256,264,270,280,288,289,294

%N Numbers n such that GCD((largest divisor of n that is <= sqrt(n)),(smallest divisor of n that is >=sqrt(n))) > 1.

%C G. Tenenbaum proved that this sequence has density 0 over the integers (théorème 3). - _Michel Marcus_, Nov 26 2012

%H Charles R Greathouse IV, <a href="/A140269/b140269.txt">Table of n, a(n) for n = 1..10000</a>

%H G. Tenenbaum, <a href="http://jlms.oxfordjournals.org/content/s2-14/3/521.full.pdf">Sur deux fonctions de diviseurs</a>, J. London Math. Soc. (1976) s2-14 (3): 521-526.

%e The divisors of 48 are 1,2,3,4,6,8,12,16,24,48. The middle two of these divisors are 6 and 8, which are not coprime. So 48 is included in this sequence.

%t okQ[n_] := (dd = Divisors[n]; s = Sqrt[n]; d1 = Select[dd, # <= s &] // Last; d2 = Select[dd, # >= s &] // First; !CoprimeQ[d1, d2]); Select[Range[300], okQ] (* _Jean-François Alcover_, Dec 27 2012 *)

%o (PARI) precdiv(n, k) = k=floor(k);while(k>0,if(n%k==0,return(k));k--)

%o in140269(n) = local(d); d=precdiv(n,sqrt(n));gcd(d,n\d)>1

%o for(n=1,300,if(in140269(n),print1(n","))) \\ _Franklin T. Adams-Watters_, Dec 20 2008

%o (PARI) is(n)=if(issquare(n),n,my(d=divisors(n));gcd(d[#d\2],d[#d\2+1]))>1 \\ _Charles R Greathouse IV_, Nov 26 2012

%Y Subsequence of A013929.

%Y Cf. A033676, A033677, A140270, A033676, A033677.

%K nonn

%O 1,1

%A _Leroy Quet_, May 16 2008

%E More terms from _Franklin T. Adams-Watters_, Dec 20 2008

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Last modified December 3 18:47 EST 2016. Contains 278745 sequences.