A324508 Numbers m such that Product_{d|m} (d/tau(d)) is an integer h where tau(k) = the number of divisors of k (A000005).

1, 2, 12, 24, 36, 72, 216, 240, 480, 720, 900, 1440, 1764, 1800, 2160, 3360, 3528, 3600, 4320, 4356, 5040, 5280, 6000, 6084, 6240, 6480, 6720, 7200, 7920, 8160, 8712, 9120, 9360, 10080, 10404, 10800, 11040, 12168, 12240, 12960, 12996, 13440, 13680, 13920

Offset

1,2

Comments

Product_{d|n} (d/tau(d)) >= 1 for all n >= 1.

Corresponding values of integers h: 1, 1, 6, 36, 216, 7776, 43046721, 2916000000, 33177600000000, ...

From Robert Israel, Mar 24 2019: (Start)

All terms other than 1 and 2 are divisible by 12.

Contains 36*p^2 for all primes p >= 5, and 480*p for all primes p >= 7. In particular, the sequence is infinite. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

A324507(a(n)) = 1.

Example

12 is a term because Product_{d|12} (d/tau(d)) = (1/tau(1)) * (2/tau(2)) * (3/tau(3)) * (4/tau(4)) * (6/tau(6)) * (12/tau(12)) = (1/1) * (2/2) * (3/2) * (4/3) * (6/4) * (12/6) = 6 (integer).

Maple

filter:= proc(n) uses numtheory; local d;
  mul(d/tau(d), d=divisors(n))::integer;
end proc:
select(filter, [$1..20000]); # Robert Israel, Mar 24 2019

Mathematica

Select[Range[15000], IntegerQ[Product[k/DivisorSigma[0, k], {k, Divisors[#]}]] &] (* G. C. Greubel, Mar 04 2019 *)

Prog

(Magma) [n: n in [1..1000] | IsIntegral(&*[d / #[c: c in Divisors(d)] : d in Divisors(n)])]
(Sage) [n for n in (1..15000) if (product(k/sigma(k, 0) for k in n.divisors())).is_integer()] # G. C. Greubel, Mar 04 2019

Crossrefs

Cf. A000005, A324506, A324507.

Sequence in context: A110821 A262983 A356078 * A100786 A141586 A141758

Adjacent sequences: A324505 A324506 A324507 * A324509 A324510 A324511

Keyword

nonn

Author

Jaroslav Krizek, Mar 03 2019

Status

approved