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A348411
Numbers whose divisors have a harmonic mean with a denominator 2.
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
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
Similar sequences: A159907, A330598.
Sequence in context: A012256 A012222 A069267 * A059270 A219085 A346142
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 17 2021
STATUS
approved