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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).
2
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
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
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Mar 03 2019
STATUS
approved