A323760 Numerator of Product_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k and tau(k) = the number of the divisors of k.

1, 1, 3, 8, 5, 27, 7, 128, 27, 125, 11, 10368, 13, 343, 3375, 131072, 17, 118098, 19, 2000000, 9261, 1331, 23, 6879707136, 625, 2197, 19683, 15059072, 29, 38443359375, 31, 2147483648, 35937, 4913, 42875, 101559956668416, 37, 6859, 59319, 10240000000000, 41

Offset: 1

Author

Jaroslav Krizek, Jan 26 2019

Keywords

• nonn
• frac

Comments

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

Examples

			For n=4; Product_{d|4} (pod(d)/tau(d)) = (pod(1)/tau(1))*(pod(2)/tau(2))*(pod(4)/tau(4)) = (1/1)*(2/2)*(8/3) = 8/3; a(4) = 8.
		

Crossrefs

Cf. A211776, A266265, A323761 (denominator).

Programs

Magma

[Numerator(&*[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]

Maple

A323760 := proc(n)
    numer(A266265(n)/A211776(n)) ;
end proc:
seq(A323760(n),n=1..20) ; # R. J. Mathar, Feb 13 2019

PARI

a(n) = my(p=1, vd); fordiv(n, d, vd = divisors(d); p *= vecprod(vd)/#vd); numerator(p); \\ Michel Marcus, Jan 27 2019

Formula

a(p) = p for primes p > 2.