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
Examples
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.
Programs
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Magma
[Numerator(&+[NumberOfDivisors(d) / SumOfDivisors(d): d in Divisors(n)]): n in [1..100]]
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Mathematica
Array[Numerator@ DivisorSum[#, Divide @@ DivisorSigma[{0, 1}, #] &] &, 63] (* Michael De Vlieger, Feb 15 2019 *) -
PARI
a(n) = numerator(sumdiv(n, d, numdiv(d)/sigma(d))); \\ Michel Marcus, Feb 13 2019
Formula
a(p) = (p+3) / gcd(p+3, p+1) for p = primes p.
Comments