A383055 Numerators of the partial sums of the reciprocals of the number of unitary divisors function (A034444).

1, 3, 2, 5, 3, 13, 15, 17, 19, 5, 11, 23, 25, 13, 27, 29, 31, 8, 17, 35, 9, 37, 39, 10, 21, 43, 45, 23, 12, 97, 101, 105, 107, 109, 111, 113, 117, 119, 121, 123, 127, 16, 33, 67, 17, 69, 71, 18, 37, 75, 19, 77, 79, 20, 81, 41, 83, 21, 43, 173, 177, 179, 181, 185

Offset: 1

Author

Amiram Eldar, Apr 15 2025

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with 1, 3/2, 2, 5/2, 3, 13/4, 15/4, 17/4, 19/4, 5, 11/2, 23/4, ...
		

References

• Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 42-43.
• Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 50.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jean-Marie De Koninck and Jacques Grah, Arithmetic functions and weighted averages, Colloquium Mathematicum, Vol. 79. No. 2 (1999), pp. 249-272.

Crossrefs

The unitary analog of A104528.

Cf. A002161, A034444, A345288, A383056 (denominators).

Similar sequences: A064608, A370898, A379513.

Programs

Mathematica

Numerator[Accumulate[1/Array[2^PrimeNu[#] &, 100]]]

PARI

list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / 2^omega(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} 1/A034444(k)).

a(n)/A383056(n) = (c/sqrt(Pi)) * n / sqrt(log(n)) + O(n / log(n)^(3/2)), where c = A345288 (De Koninck and Ivić, 1980).