A379359 Numerators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

1, 2, 3, 7, 9, 11, 13, 41, 22, 25, 28, 59, 65, 71, 77, 391, 421, 218, 233, 481, 511, 541, 571, 581, 298, 313, 106, 217, 227, 237, 247, 1739, 1809, 1879, 1949, 3933, 4073, 4213, 4353, 13199, 13619, 14039, 14459, 14669, 14879, 15299, 15719, 15803, 16013, 16223, 16643

Offset: 1

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with 1, 2, 3, 7/2, 9/2, 11/2, 13/2, 41/6, 22/3, 25/3, 28/3, 59/6, ...
		

References

• Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 13-16, Theorem 1.3.
• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See section 5.1, Abelian group enumeration constants, p. 274.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Crossrefs

Cf. A000688, A063966, A084911, A370897, A379360 (denominators), A379361.

Programs

Mathematica

Numerator[Accumulate[Table[1/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

PARI

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / f(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} 1/A000688(k)).

a(n)/A379360(n) = D * n + O(sqrt(n/log(n))), where D = A084911.