A337103 Numbers k with a divisor pair (d,k/d) whose harmonic mean is an integer.

1, 4, 9, 12, 16, 18, 25, 36, 45, 48, 49, 64, 72, 81, 100, 108, 112, 121, 144, 150, 162, 169, 180, 192, 196, 225, 240, 256, 288, 289, 294, 300, 324, 361, 396, 400, 405, 432, 441, 448, 450, 484, 490, 525, 529, 576, 588, 600, 625, 637, 648, 676, 720, 729, 768, 784, 841, 882, 900, 960, 961

Offset

1,2

Comments

All positive squares are in the sequence since they have a divisor pair such that (d,k/d) = (d,d). The harmonic mean is then an integer since we have 2*d*d/(d+d) = 2*d*d/(2*d) = d.

Links

Table of n, a(n) for n=1..61.

Example

18 is in the sequence since it has the divisor pair (3,6) with harmonic mean 2*3*6/(3+6) = 36/9 = 4 (an integer).
25 is in the sequence since it has the divisor pair (5,5) with harmonic mean 2*5*5/(5+5) = 50/10 = 5 (an integer).

Mathematica

seqQ[n_] := Module[{d = Select[Divisors[n], #^2 <= n &]}, AnyTrue[d, IntegerQ @ HarmonicMean[{#, n/#}] &]]; Select[Range[1000], seqQ] (* Amiram Eldar, Aug 18 2020 *)

Prog

(PARI) isok(k) = {fordiv(k, d, if (denominator(2*k*d/(d^2+k)) == 1, return (1)); ); } \\ Michel Marcus, Aug 16 2020

Crossrefs

Cf. A000290.

Sequence in context: A351575 A348739 A135572 * A312855 A348752 A365869

Adjacent sequences: A337100 A337101 A337102 * A337104 A337105 A337106

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 16 2020

Status

approved