A286325 - OEIS

%I #17 Mar 24 2023 08:40:02

%S 1,6,45,60,90,270,420,630,672,2970,5460,8190,9072,9100,10080,15925,

%T 22680,22848,27300,30240,40950,45360,54600,81900,95550,99792,136500,

%U 163800,172900,204750,208656,245700,249480,312480,332640,342720,385560,409500,472500,491400

%N Bi-unitary harmonic numbers.

%C A number m is a term if the sum of its bi-unitary divisors, A188999(m) divides the product of m by the number of its bi-unitary divisors A286324(m).

%C Numbers k whose harmonic mean of their bi-unitary divisors, A361782(k)/A361783(k), is an integer. - _Amiram Eldar_, Mar 24 2023

%H Amiram Eldar, <a href="/A286325/b286325.txt">Table of n, a(n) for n = 1..313</a> (all terms below 10^10, first 40 terms from Michel Marcus)

%H Jozsef Sandor, <a href="https://arxiv.org/abs/1105.0294">On bi-unitary harmonic numbers</a>, arXiv:1105.0294 [math.NT], 2011. See pp. 13-20, but beware of typos and possible errors.

%t f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; bhQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; bhQ[1] = True; Select[Range[10^5], bhQ] (* _Amiram Eldar_, Mar 24 2023 *)

%o (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

%o gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

%o biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));

%o isok(n) = my(v=biudivs(n)); denominator(n*#v/vecsum(v))==1;

%Y Cf. A222266, A188999, A286324, A361782, A361783, A361784.

%Y Cf. A001599 (Ore harmonic), A006086 (unitary harmonic).

%K nonn

%O 1,2

%A _Michel Marcus_, May 07 2017