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

Author

Amiram Eldar, Oct 17 2021

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A001599, A099377, A099378, A348412.

Similar sequences: A159907, A330598.

Programs

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 *)

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