login
Numbers that are an integer multiple of the mean of their smallest and largest nontrivial divisors.
1

%I #14 Jul 24 2017 20:02:17

%S 4,9,12,25,45,49,121,169,289,361,529,637,841,961,1369,1681,1849,2209,

%T 2809,3481,3721,4489,5041,5329,6241,6889,7921,9409,10201,10609,11449,

%U 11881,12769,13357,16129,17161,18769,19321,22201,22801,24649,26569,27889,29929,32041,32761,36481,37249,38809,39601,44521

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

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

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

%C 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.

%H Harvey P. Dale, <a href="/A283105/b283105.txt">Table of n, a(n) for n = 1..600</a>

%e 4 is in the sequence because its smallest nontrivial divisor is 2, its largest nontrivial divisor is 2, and their mean is 2.

%e 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.

%e 10 is not in the sequence because it is not an integral multiple of 7/2, the mean of 2 and 5.

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

%o (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

%Y Cf. A005382, A088595.

%K nonn

%O 1,1

%A _Emmanuel Vantieghem_, Feb 28 2017