A323781 Numbers m such that Sum_{d|m} (tau(d)/sigma(d)) is an integer h where tau(k) = the number of the divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

1, 15, 429, 609, 6003, 9156, 20943, 75579, 90252, 93849, 115773, 331359, 631764, 744993, 817191, 837655, 925083, 1130766, 1141191, 2349087, 2491740, 2512965, 3040728, 3266253, 3796143, 4314891, 4365231, 5025930, 5294340, 6135624, 6629271, 7210671, 10906175

Offset: 1

Author

Jaroslav Krizek, Feb 16 2019

Keywords

• nonn

Comments

Sum_{d|n} (tau(d)/sigma(d)) > 1 for all n > 2.

Corresponding values of integers h: 1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 5, 2, 2, 2, 2, 4, 2, 2, 5, 3, 4, 2, 2, 2, 2, 5, 5, 5, 2, 2, 2, ...

The smallest number m such that Sum_{d|m} (tau(d)/sigma(d)) is an integer h for h >= 1: 1, 15, 2512965, 9156, 631764, ...

Examples

			15 is a term because Sum_{d|15} (tau(d)/sigma(d)) = tau(1)/sigma(1) + tau(3)/sigma(3) + tau(5)/sigma(5) + tau(15)/sigma(15) = 1/1 + 2/4 + 2/6 + 4/24 = 2 (integer).
		

Crossrefs

Cf. A000005, A000203, A323779, A323780.

Programs

Magma

[n: n in [1..1000000] | Denominator(&+[NumberOfDivisors(d) / SumOfDivisors(d): d in Divisors(n)]) eq 1]

Mathematica

Select[Range[10^5], IntegerQ@ DivisorSum[#, Divide @@ DivisorSigma[{0, 1}, #] &] &] (* Michael De Vlieger, Feb 17 2019 *)

PARI

isok(n) = !frac(sumdiv(n, d, numdiv(d)/sigma(d))); \\ Michel Marcus, Feb 16 2019

Formula

A323780(a(n)) = 1.