A323780 a(n) = denominator 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, 3, 2, 21, 3, 2, 4, 105, 26, 9, 6, 7, 7, 12, 1, 3255, 9, 26, 10, 63, 8, 18, 12, 35, 93, 21, 65, 21, 15, 3, 16, 1085, 4, 27, 3, 91, 19, 6, 7, 315, 21, 8, 22, 9, 13, 36, 24, 2170, 76, 279, 3, 147, 27, 39, 9, 21, 20, 9, 30, 21, 31, 48, 104, 137795, 21, 12, 34

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..67.

Formula

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

a(n) = 1 for numbers in A323781.

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) = 21.

Mathematica

Array[Denominator@ DivisorSum[#, Divide @@ DivisorSigma[{0, 1}, #] &] &, 67] (* Michael De Vlieger, Feb 15 2019 *)

Prog

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

Crossrefs

Cf. A000005, A000203, A323779 (numerator), A323781.

Sequence in context: A009022 A009033 A298661 * A350290 A248123 A018872

Adjacent sequences: A323777 A323778 A323779 * A323781 A323782 A323783

Keyword

nonn,frac

Author

Jaroslav Krizek, Feb 13 2019

Status

approved