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

1,3

Comments

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..7500

Formula

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

Example

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.

Maple

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

Mathematica

A323760[n_] := Numerator[Times @@ Map[Times @@ #/Length[#] &, Divisors[Rest[Divisors[n]]]]];
Array[A323760, 50] (* Paolo Xausa, Sep 11 2025 *)

Prog

(Magma) [Numerator(&*[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]];
(PARI) a(n) = my(p=1, vd); fordiv(n, d, vd = divisors(d); p *= vecprod(vd)/#vd); numerator(p); \\ Michel Marcus, Jan 27 2019

Crossrefs

Cf. A211776, A266265, A323761 (denominator).

Sequence in context: A086872 A389125 A363421 * A054792 A144872 A090347

Adjacent sequences: A323757 A323758 A323759 * A323761 A323762 A323763

Keyword

nonn,frac

Author

Jaroslav Krizek, Jan 26 2019

Status

approved