A370897 -id:A370897 - cp's OEIS frontend

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

Amiram Eldar, Dec 21 2024

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

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.

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.

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

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

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

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

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

A379360 Denominators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 30, 30, 15, 15, 30, 30, 30, 30, 30, 15, 15, 5, 10, 10, 10, 10, 70, 70, 70, 70, 140, 140, 140, 140, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 140, 140, 140, 140, 140, 140, 140, 140

Offset: 1

Amiram Eldar, Dec 21 2024

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.

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.

Cf. A000688, A063966, A370897, A379359 (numerators), A379362.

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

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

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

A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 3, 7, 5, 2, 5, 7, 13, 7, 13, 59, 89, 37, 52, 89, 119, 89, 119, 109, 62, 47, 52, 89, 119, 89, 119, 803, 1013, 803, 1013, 1921, 2341, 1921, 2341, 2201, 2621, 2201, 2621, 2411, 2621, 2201, 2621, 2537, 2747, 2537, 2957, 2747, 3167, 1009, 1149, 3307

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 0, 1, 1/2, 3/2, 1/2, 3/2, 7/6, 5/3, 2/3, 5/3, 7/6, ...

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.

Cf. A000041, A000688, A063966, A084911, A370897, A379359, A379362 (denominators).

Numerator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

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

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

a(n)/A379362(n) ~ D * c * n, where D = A084911, c = 2/(1 + Sum_{k>=1} 1/(P(k)*2^k)) - 1 = 0.18634377034863729099..., and P(k) = A000041(k).

A379362 Denominators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 30, 30, 15, 15, 30, 30, 30, 30, 30, 15, 15, 15, 30, 30, 30, 30, 210, 210, 210, 210, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 140, 140, 420, 420, 420, 420, 420, 420, 420

Offset: 1

Amiram Eldar, Dec 21 2024

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.

Cf. A000688, A063966, A370897, A379360, A379361 (numerators).

Denominator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

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

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).

cp's OEIS Frontend

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

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A379360 Denominators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

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A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

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Mathematica

PARI

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A379362 Denominators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula