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

1,1

Comments

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

Links

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

Example

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.

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[#]]]]])&]

Prog

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

Crossrefs

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

Sequence in context: A246131 A068999 A179707 * A077438 A350343 A001248

Adjacent sequences: A247075 A247076 A247077 * A247079 A247080 A247081

Keyword

nonn

Author

Daniel Lignon, Nov 17 2014

Status

approved