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

1,2

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.

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.

Example

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

Mathematica

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

Prog

(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), ", "))};

Crossrefs

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

Sequence in context: A345745 A280110 A203234 * A212985 A103797 A262814

Adjacent sequences: A379356 A379357 A379358 * A379360 A379361 A379362

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 21 2024

Status

approved