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

1,1

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.

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A005382, A088595.

Sequence in context: A076794 A254520 A344578 * A179808 A083351 A055381

Adjacent sequences: A283102 A283103 A283104 * A283106 A283107 A283108

Keyword

nonn

Author

Emmanuel Vantieghem, Feb 28 2017

Status

approved