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).

1, 2, 12, 18, 24, 36, 54, 60, 72, 84, 90, 108, 120, 126, 132, 150, 156, 168, 180, 198, 204, 216, 228, 234, 240, 252, 264, 270, 276, 294, 300, 306, 312, 342, 348, 360, 372, 378, 396, 408, 414, 420, 444, 450, 456, 468, 480, 492, 504, 516, 522, 540, 552, 558, 564

Offset: 1

Author

Jaroslav Krizek, Jan 27 2019

Keywords

• nonn

Comments

Corresponding values of integers h: 1, 1, 10368, 118098, 6879707136, 101559956668416, ...

Product_{d|n} (pod(d)/tau(d)) > 1 for all n > 2.

Examples

			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).
		

Crossrefs

Cf. A000005, A007955, A323760, A323761.

Programs

Magma

[n: n in [1..1000] | Denominator(&*[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]) eq 1]

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

Formula

A323761(a(n)) = 1.