

A247078


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


3



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
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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

Daniel Lignon, Table of n, a(n) for n = 1..1269


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((#d2)/sum(i=2, #d1, 1/d[i])) == 1);


CROSSREFS

Cf. similar sequences: A001599 (with all divisors), A247077 (with proper divisors).
Sequence in context: A246131 A068999 A179707 * A077438 A001248 A280076
Adjacent sequences: A247075 A247076 A247077 * A247079 A247080 A247081


KEYWORD

nonn


AUTHOR

Daniel Lignon, Nov 17 2014


STATUS

approved



