A247078 Numbers for which the harmonic mean of nontrivial divisors is an integer.

4, 9, 25, 49, 121, 169, 289, 345, 361, 529, 841, 961, 1050, 1369, 1645, 1681, 1849, 2209, 2809, 3481, 3721, 4386, 4489, 5041, 5329, 6241, 6489, 6889, 7921, 8041, 9409, 10201, 10609, 11449, 11881, 12769, 13026, 16129, 17161, 18769, 19321, 22201, 22801

Offset: 1

Author

Daniel Lignon, Nov 17 2014

Keywords

• nonn

Comments

All the squares of prime numbers (A001248) have this property but there are other numbers (A247079): 345, 1050, 1645, 4386, 6489, 8041, ...

Examples

			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.
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1269 from Daniel Lignon)

Crossrefs

Cf. similar sequences: A001599 (with all divisors), A247077 (with proper divisors).

Programs

Maple

hm:= S -> nops(S)/convert(map(t->1/t,S),`+`):
filter:= n -> not isprime(n) and type(hm(numtheory:-divisors(n) minus {1,n}),integer):
select(filter, [$2..10^5]); # Robert Israel, Nov 17 2014

Mathematica

Select[Range[2,100000],(Not[PrimeQ[#]] && IntegerQ[HarmonicMean[Rest[Most[Divisors[#]]]]])&]

PARI

isok(n) = my(d=divisors(n)); (#d >2) && (denominator((#d-2)/sum(i=2, #d-1, 1/d[i])) == 1);