cp's OEIS Frontend

A323762 Numbers m such that Product_{d|m} (pod(d)/tau(d)) is an integer h where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).

%I A323762 #15 Dec 15 2024 07:58:11
%S A323762 1,2,12,18,24,36,54,60,72,84,90,108,120,126,132,150,156,168,180,198,
%T A323762 204,216,228,234,240,252,264,270,276,294,300,306,312,342,348,360,372,
%U A323762 378,396,408,414,420,444,450,456,468,480,492,504,516,522,540,552,558,564
%N A323762 Numbers m such that Product_{d|m} (pod(d)/tau(d)) is an integer h where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).
%C A323762 Corresponding values of integers h: 1, 1, 10368, 118098, 6879707136, 101559956668416, ...
%C A323762 Product_{d|n} (pod(d)/tau(d)) > 1 for all n > 2.
%F A323762 A323761(a(n)) = 1.
%e A323762 12 is a term because Product_{d|12} (pod(d)/tau(d)) = (pod(1)/tau(1))*(pod(2)/tau(2))*(pod(3)/tau(3))*(pod(4)/tau(4))*(pod(6)/tau(6))*(pod(12)/tau(12)) = (1/1)*(2/2)*(3/2)*(8/3)*(36/4)*(1728/6) = 10368 (integer).
%o A323762 (Magma) [n: n in [1..1000] | Denominator(&*[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]) eq 1]
%o A323762 (PARI) isok(n) = my(p=1, vd); fordiv(n, d, vd = divisors(d); p *= vecprod(vd)/#vd); denominator(p) == 1; \\ _Michel Marcus_, Jan 30 2019
%Y A323762 Cf. A000005, A007955, A323760, A323761.
%K A323762 nonn
%O A323762 1,2
%A A323762 _Jaroslav Krizek_, Jan 27 2019