A324509 a(n) = numerator of Product_{d|n} (sigma(d)/tau(d)) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

1, 3, 2, 7, 3, 9, 4, 105, 26, 81, 6, 98, 7, 36, 36, 651, 9, 507, 10, 1323, 64, 81, 12, 11025, 31, 441, 260, 784, 15, 6561, 16, 13671, 144, 729, 144, 753571, 19, 225, 196, 893025, 21, 20736, 22, 2646, 2028, 324, 24, 423801, 76, 25947, 324, 16807, 27, 38025, 324

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..55.

Formula

a(p) = (p+1)/2 for odd primes p.

Example

Product_{d|n} (sigma(d)/tau(d)) for n >= 1: 1, 3/2, 2, 7/2, 3, 9, 4, 105/8, 26/3, 81/4, 6, 98, 7, 36, 36, 651/8, ...
For n=4; Product_{d|4} (sigma(d)/tau(d)) = sigma(1)/tau(1) + sigma(2)/tau(2) + sigma(4)/tau(4) = (1/1) * (3/2) * (7/3) = 7/2; a(4) = 7.

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A000203, A324510 (denominators).

Sequence in context: A257322 A276466 A289336 * A335653 A296512 A241558

Adjacent sequences: A324506 A324507 A324508 * A324510 A324511 A324512

Keyword

nonn,frac

Author

Jaroslav Krizek, Mar 03 2019

Status

approved