A324507 a(n) = denominator of Product_{d|n} (d/tau(d)) where tau(k) = the number of divisors of k (A000005).

1, 1, 2, 3, 2, 4, 2, 3, 2, 4, 2, 1, 2, 4, 16, 15, 2, 4, 2, 9, 16, 4, 2, 1, 6, 4, 8, 9, 2, 256, 2, 45, 16, 4, 16, 1, 2, 4, 16, 9, 2, 256, 2, 9, 32, 4, 2, 25, 6, 36, 16, 9, 2, 64, 16, 9, 16, 4, 2, 16, 2, 4, 32, 315, 16, 256, 2, 9, 16, 256, 2, 1, 2, 4, 32, 9, 16

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) = 2 for odd primes p.

a(n) = 1 for numbers n in A324508.

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) = 3.

Maple

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

Mathematica

Table[Denominator[Product[k/DivisorSigma[0, k], {k, Divisors[n]}]], {n, 1, 100}] (* G. C. Greubel, Mar 04 2019 *)

Prog

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

Crossrefs

Cf. A000005, A324506 (numerators), A324508.

Sequence in context: A340218 A199968 A066482 * A123725 A089080 A085058

Adjacent sequences: A324504 A324505 A324506 * A324508 A324509 A324510

Keyword

nonn,frac

Author

Jaroslav Krizek, Mar 03 2019

Status

approved