A283105 Numbers that are an integer multiple of the mean of their smallest and largest nontrivial divisors.

4, 9, 12, 25, 45, 49, 121, 169, 289, 361, 529, 637, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 13357, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521

Offset: 1

Author

Emmanuel Vantieghem, Feb 28 2017

Keywords

• nonn

Comments

No prime is in the sequence since there are no nontrivial divisors of a prime.

The sequence includes every number that is the square of a prime.

It is easy to show that the other terms are of the form (2p-1)*p^2 where p and 2p-1 are prime. Therefore, the mean of the two divisors in question is always an integer.

Examples

			4 is in the sequence because its smallest nontrivial divisor is 2, its largest nontrivial divisor is 2, and their mean is 2.
45 is in the sequence because its smallest nontrivial divisor is 3, its largest nontrivial divisor is 15, and their mean is 9, a divisor of 45.
10 is not in the sequence because it is not an integral multiple of 7/2, the mean of 2 and 5.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..600

Crossrefs

Cf. A005382, A088595.

Programs

Mathematica

mslndQ[n_]:=Module[{d=Divisors[n]},Divisible[n,Mean[{d[[2]],d[[-2]]}]]]; Select[Range[2,50000],mslndQ] (* Harvey P. Dale, Jul 24 2017 *)

PARI

is(n) = my(d=divisors(n), m=(d[2]+d[#d-1])/2); if(n%m==0, 1, 0) \\ Felix Fröhlich, Feb 28 2017