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

4, 8, 9, 16, 18, 24, 25, 27, 32, 36, 48, 49, 50, 54, 60, 64, 75, 80, 81, 96, 98, 100, 108, 112, 120, 121, 125, 128, 135, 140, 144, 147, 150, 160, 162, 168, 169, 180, 189, 192, 196, 200, 216, 224, 225, 242, 243, 245, 250, 252, 256, 264, 270, 280, 288, 289, 294

Offset

1,1

Comments

G. Tenenbaum proved that this sequence has density 0 over the integers (théorème 3). - Michel Marcus, Nov 26 2012

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

G. Tenenbaum, Sur deux fonctions de diviseurs, J. London Math. Soc. (1976) s2-14 (3): 521-526.

Example

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.

Mathematica

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 *)

Prog

(PARI) precdiv(n, k) = k=floor(k); while(k>0, if(n%k==0, return(k)); k--)
in140269(n) = local(d); d=precdiv(n, sqrt(n)); gcd(d, n\d)>1
for(n=1, 300, if(in140269(n), print1(n", "))) \\ Franklin T. Adams-Watters, Dec 20 2008
(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

Crossrefs

Subsequence of A013929.

Cf. A033676, A033677, A140270, A033676, A033677.

Sequence in context: A034024 A299117 A339462 * A339744 A226385 A070003

Adjacent sequences: A140266 A140267 A140268 * A140270 A140271 A140272

Keyword

nonn

Author

Leroy Quet, May 16 2008

Extensions

More terms from Franklin T. Adams-Watters, Dec 20 2008

Status

approved