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%I #20 Sep 08 2022 08:46:24
%S 1,2,12,24,36,72,216,240,480,720,900,1440,1764,1800,2160,3360,3528,
%T 3600,4320,4356,5040,5280,6000,6084,6240,6480,6720,7200,7920,8160,
%U 8712,9120,9360,10080,10404,10800,11040,12168,12240,12960,12996,13440,13680,13920
%N Numbers m such that Product_{d|m} (d/tau(d)) is an integer h where tau(k) = the number of divisors of k (A000005).
%C Product_{d|n} (d/tau(d)) >= 1 for all n >= 1.
%C Corresponding values of integers h: 1, 1, 6, 36, 216, 7776, 43046721, 2916000000, 33177600000000, ...
%C From _Robert Israel_, Mar 24 2019: (Start)
%C All terms other than 1 and 2 are divisible by 12.
%C Contains 36*p^2 for all primes p >= 5, and 480*p for all primes p >= 7. In particular, the sequence is infinite. (End)
%H Robert Israel, <a href="/A324508/b324508.txt">Table of n, a(n) for n = 1..10000</a>
%F A324507(a(n)) = 1.
%e 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).
%p filter:= proc(n) uses numtheory; local d;
%p mul(d/tau(d), d=divisors(n))::integer;
%p end proc:
%p select(filter, [$1..20000]); # _Robert Israel_, Mar 24 2019
%t Select[Range[15000], IntegerQ[Product[k/DivisorSigma[0, k], {k, Divisors[#]}]] &] (* _G. C. Greubel_, Mar 04 2019 *)
%o (Magma) [n: n in [1..1000] | IsIntegral(&*[d / #[c: c in Divisors(d)] : d in Divisors(n)])]
%o (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
%Y Cf. A000005, A324506, A324507.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Mar 03 2019
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