A348411 Numbers whose divisors have a harmonic mean with a denominator 2.

3, 15, 42, 84, 135, 308, 420, 546, 1428, 1488, 1890, 2295, 2660, 3780, 6210, 7440, 9424, 12180, 13392, 18018, 20832, 24384, 24570, 43152, 43400, 64260, 66960, 77490, 90090, 98420, 121920, 127710, 155610, 200340, 204600, 227664, 316992, 348688, 353400, 461776, 483210

Offset

1,1

Comments

Numbers k such that A099378(k) = 2.

The odd terms seem to be relatively rare: 3, 15, 135, 2295, 544635, 9258795, 22330035, 39118408875, ...

If k is in this sequence, then 2*k is in A348412.

Links

Amiram Eldar, Table of n, a(n) for n = 1..310

Example

3 is a term since the harmonic mean of its divisors, {1, 3}, is 3/2.
15 is a term since the harmonic mean of its divisors, {1, 3, 5, 15}, is 5/2.

Maple

filter:= proc(n) local L, h;
  L:= map(t->1/t, numtheory:-divisors(n));
  denom(nops(L)/convert(L, `+`))=2;
end proc:
select(filter, [$1..10^6]); # Robert Israel, Oct 17 2021

Mathematica

Select[Range[10^5], Denominator[DivisorSigma[0, #]/DivisorSigma[-1, #]] == 2 &]
Select[Range[500000], Denominator[HarmonicMean[Divisors[#]]]==2&] (* Harvey P. Dale, Apr 06 2023 *)

Prog

(PARI) isok(m) = my(d=divisors(m)); denominator(#d/sum(k=1, #d, 1/d[k])) == 2; \\ Michel Marcus, Oct 18 2021
(Python)
from sympy import gcd, divisor_sigma
A348411_list = [n for n in range(1, 10**3) if (lambda x, y: 2*gcd(x, y*n)==x)(divisor_sigma(n), divisor_sigma(n, 0))] # Chai Wah Wu, Oct 20 2021

Crossrefs

Cf. A001599, A099377, A099378, A348412.

Similar sequences: A159907, A330598.

Sequence in context: A012256 A012222 A069267 * A059270 A219085 A346142

Adjacent sequences: A348408 A348409 A348410 * A348412 A348413 A348414

Keyword

nonn

Author

Amiram Eldar, Oct 17 2021

Status

approved