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Numbers for which the harmonic mean of nontrivial divisors is an integer.
5

%I #47 Nov 10 2024 16:15:01

%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,changed

%O 1,1

%A _Daniel Lignon_, Nov 17 2014