

A337103


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


0



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
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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.


FORMULA

Characteristic function: sign( Sum_{dn} chi(2*n*d/(n+d^2)) ), where chi is the integer characteristic.


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: A240112 A162645 A135572 * A312855 A312856 A010449
Adjacent sequences: A337100 A337101 A337102 * A337104 A337105 A337106


KEYWORD

nonn


AUTHOR

Wesley Ivan Hurt, Aug 16 2020


STATUS

approved



