A286325 Bi-unitary harmonic numbers.

1, 6, 45, 60, 90, 270, 420, 630, 672, 2970, 5460, 8190, 9072, 9100, 10080, 15925, 22680, 22848, 27300, 30240, 40950, 45360, 54600, 81900, 95550, 99792, 136500, 163800, 172900, 204750, 208656, 245700, 249480, 312480, 332640, 342720, 385560, 409500, 472500, 491400

Offset

1,2

Comments

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).

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..313 (all terms below 10^10, first 40 terms from Michel Marcus)

Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011. See pp. 13-20, but beware of typos and possible errors.

Mathematica

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 *)

Prog

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
isok(n) = my(v=biudivs(n)); denominator(n*#v/vecsum(v))==1;

Crossrefs

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

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

Sequence in context: A077672 A119202 A367561 * A335387 A063947 A006086

Adjacent sequences: A286322 A286323 A286324 * A286326 A286327 A286328

Keyword

nonn

Author

Michel Marcus, May 07 2017

Status

approved