A323779 a(n) = numerator of Sum_{d|n} (tau(d)/sigma(d)) where tau(k) = the number of the divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

1, 5, 3, 44, 4, 5, 5, 248, 45, 20, 7, 22, 8, 25, 2, 8213, 10, 75, 11, 176, 15, 35, 13, 124, 133, 40, 119, 55, 16, 10, 17, 2841, 7, 50, 5, 330, 20, 11, 12, 992, 22, 25, 23, 22, 30, 65, 25, 8213, 99, 665, 5, 352, 28, 119, 14, 62, 33, 16, 31, 88, 32, 85, 225

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..63.

Formula

a(p) = (p+3) / gcd(p+3, p+1) for p = primes p.

Example

For n=4; Sum_{d|4} (tau(d)/sigma(d)) = (tau(1)/sigma(1))+(tau(2)/sigma(2))+(tau(4)/sigma(4)) = (1/1)+(2/3)+(3/7) = 44/21; a(4) = 44.

Mathematica

Array[Numerator@ DivisorSum[#, Divide @@ DivisorSigma[{0, 1}, #] &] &, 63] (* Michael De Vlieger, Feb 15 2019 *)

Prog

(Magma) [Numerator(&+[NumberOfDivisors(d) / SumOfDivisors(d): d in Divisors(n)]): n in [1..100]]
(PARI) a(n) = numerator(sumdiv(n, d, numdiv(d)/sigma(d))); \\ Michel Marcus, Feb 13 2019

Crossrefs

Cf. A000005, A000203, A323780 (denominator).

Sequence in context: A291843 A187278 A288184 * A343290 A027858 A181755

Adjacent sequences: A323776 A323777 A323778 * A323780 A323781 A323782

Keyword

nonn,frac

Author

Jaroslav Krizek, Jan 27 2019

Status

approved