A324506 a(n) = numerator of Product_{d|n} (d/tau(d)) where tau(k) = the number of divisors of k (A000005).

1, 1, 3, 4, 5, 9, 7, 8, 9, 25, 11, 6, 13, 49, 225, 128, 17, 81, 19, 250, 441, 121, 23, 36, 125, 169, 243, 686, 29, 50625, 31, 2048, 1089, 289, 1225, 216, 37, 361, 1521, 2500, 41, 194481, 43, 2662, 10125, 529, 47, 13824, 343, 15625, 2601, 4394, 53, 59049, 3025

Offset

1,3

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(p) = p for p = odd primes.

Example

Product_{d|n} (d/tau(d)) for n >= 1: 1, 1, 3/2, 4/3, 5/2, 9/4, 7/2, 8/3, 9/2, 25/4, 11/2, 6, 13/2, 49/4, 225/16, ...
For n=4; Product_{d|4} (d/tau(d)) = (1/tau(1)) * (2/tau(2)) * (4/tau(4)) = (1/1) * (2/2) * (4/3) = 4/3; a(4) = 4.

Maple

f:= proc(n) local d; numer(mul(d/numtheory:-tau(d), d=numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 04 2021

Mathematica

Table[Numerator[Product[k/DivisorSigma[0, k], {k, Divisors[n]}]], {n, 1, 60}] (* G. C. Greubel, Mar 04 2019 *)

Prog

(Magma) [Numerator(&*[d / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
(Sage) [product(k/sigma(k, 0) for k in n.divisors()).numerator() for n in (1..60)] # G. C. Greubel, Mar 04 2019

Crossrefs

Cf. A000005, A324507 (denominators).

Sequence in context: A349659 A302653 A300738 * A342772 A065336 A079097

Adjacent sequences: A324503 A324504 A324505 * A324507 A324508 A324509

Keyword

nonn,frac

Author

Jaroslav Krizek, Mar 03 2019

Status

approved