|
%I #43 Mar 03 2020 09:00:49
%S 4,9,25,49,121,169,289,345,361,529,841,961,1050,1369,1645,1681,1849,
%T 2209,2809,3481,3721,4386,4489,5041,5329,6241,6489,6889,7921,8041,
%U 9409,10201,10609,11449,11881,12769,13026,16129,17161,18769,19321,22201,22801
%N Numbers for which the harmonic mean of nontrivial divisors is an integer.
%C All the squares of prime numbers (A001248) have this property but there are other numbers (A247079): 345, 1050, 1645, 4386, 6489, 8041, ...
%H Amiram Eldar, <a href="/A247078/b247078.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1269 from Daniel Lignon)
%e The divisors of 25 are [1,5,25] and the nontrivial divisors are [5]. The harmonic mean is 1/(1/5))=5. That's the same for all squares of prime numbers.
%e The nontrivial divisors of 345 are [3,5,15,23,69,115] and their harmonic mean is 6/(1/3+1/5+1/15+1/23+1/69+1/115) = 9.
%p hm:= S -> nops(S)/convert(map(t->1/t,S),`+`):
%p filter:= n -> not isprime(n) and type(hm(numtheory:-divisors(n) minus {1,n}),integer):
%p select(filter, [$2..10^5]); # _Robert Israel_, Nov 17 2014
%t Select[Range[2,100000],(Not[PrimeQ[#]] && IntegerQ[HarmonicMean[Rest[Most[Divisors[#]]]]])&]
%o (PARI) isok(n) = my(d=divisors(n)); (#d >2) && (denominator((#d-2)/sum(i=2, #d-1, 1/d[i])) == 1);
%Y Cf. similar sequences: A001599 (with all divisors), A247077 (with proper divisors).
%K nonn
%O 1,1
%A _Daniel Lignon_, Nov 17 2014
|
