A104529 -id:A104529 - cp's OEIS frontend

A104528 Numerator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.

1, 3, 2, 7, 17, 37, 43, 23, 25, 53, 59, 61, 67, 35, 73, 377, 407, 139, 149, 457, 118, 487, 517, 1049, 363, 373, 383, 1169, 1229, 311, 163, 331, 677, 173, 707, 2141, 2231, 569, 2321, 4687, 4867, 614, 1273, 644, 1303, 2651, 2741, 2759, 2819, 2849, 1447, 731

Offset: 1

Emeric Deutsch, Mar 12 2005

			Fractions begin with 1, 3/2, 2, 7/3, 17/6, 37/12, 43/12, 23/6, 25/6, 53/12, 59/12, 61/12, ...
a(4) = 7 because 1/tau(1) + 1/tau(2) + 1/tau(3) + 1/tau(4) = 1/1 + 1/2 + 1/2 + 1/3 = 7/3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Srinivasan Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Math., Vol. 45 (1916), pp. 81-84.
B. M. Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, Volume s2-21, Issue 1 (1923), pp. 235-255.

Cf. A000005, A104529 (denominators).

with(numtheory): a:=n->numer(sum(1/tau(k),k=1..n)): seq(a(n),n=1..57);

Numerator[Accumulate[1/Array[DivisorSigma[0, #] &, 50]]] (* Amiram Eldar, Oct 14 2022 *)

Sum_{k=1..n} a(k)/A104529(k) ~ (n/sqrt(log(n))) * (c_0 + c_1/log(n) + ... + c_k/log(n)^k + O(1/log(n)^(k+1))), where c_0 = (1/sqrt(Pi)) * Product_{p prime} sqrt(p^2-p) * log(p/(p-1)) (Ramanujan, 1916; Wilson, 1923). - Amiram Eldar, Oct 14 2022

A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

Original entry on oeis.org

1, 2, 1, 3, 6, 12, 12, 6, 2, 4, 4, 12, 12, 6, 12, 60, 60, 60, 60, 20, 5, 20, 20, 40, 120, 120, 120, 120, 120, 15, 30, 5, 20, 5, 20, 180, 180, 90, 180, 360, 360, 45, 90, 45, 90, 180, 180, 180, 180, 180, 45, 90, 45, 360, 360, 45, 180, 45, 90, 180, 180, 45, 90, 630

Offset: 1

Amiram Eldar, Oct 16 2022

See A357843 for more details.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000005, A307704, A357843 (numerators).

Similar sequences: A104529, A211178, A357821.

Denominator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]

lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / numdiv(k); print1(denominator(s), ", "))};

from fractions import Fraction
from sympy import divisor_count
def A357844(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, Oct 16 2022

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/d(k)), where d(k) = A000005(k).

A357846 Denominators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 12, 84, 84, 7, 56, 840, 10920, 32760, 32760, 32760, 32760, 16380, 32760, 1015560, 338520, 338520, 338520, 338520, 1354080, 4062240, 4062240, 4062240, 131040, 131040, 131040, 131040, 131040, 43680, 21840, 65520, 32760, 98280, 196560, 196560, 3734640, 3734640

Offset: 1

Amiram Eldar, Oct 16 2022

See A357845 for more details.

Olivier Bordellès and Benoit Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000203, A068762, A357845 (numerators).

Similar sequence: A104529, A212718, A357821.

Denominator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[1, #] &, 60]]]

lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / sigma(k); print1(denominator(s), ", "))};

from fractions import Fraction
from sympy import divisor_sigma
def A357846(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, Oct 16 2022

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/sigma(k)), where sigma(k) = A000203(k).

A379358 Denominators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

Original entry on oeis.org

1, 3, 3, 6, 6, 18, 18, 45, 90, 90, 90, 45, 45, 45, 15, 1, 3, 18, 18, 9, 9, 1, 3, 30, 15, 45, 90, 5, 15, 135, 135, 945, 945, 945, 945, 3780, 3780, 3780, 3780, 756, 756, 756, 756, 756, 756, 756, 756, 3780, 3780, 3780, 3780, 3780, 3780, 756, 756, 3780, 3780, 3780

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. 12-13, Theorem 1.2.
József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the asymptotic formulae for some functions connected with powers of the zeta-function, Matematički Vesnik, Vol. 1 (14) (29) (1977), pp. 79-90.

Cf. A007425, A061201, A104529, A379357 (numerators).

f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/d3[n], {n, 1, 100}]]]

d3(n) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / d3(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A007425(k)).

A383056 Denominators of the partial sums of the reciprocals of the number of unitary divisors function (A034444).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 4, 4, 4, 1, 2, 4, 4, 2, 4, 4, 4, 1, 2, 4, 1, 4, 4, 1, 2, 4, 4, 2, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 2, 4, 1, 4, 4, 1, 2, 4, 1, 4, 4, 1, 4, 2, 4, 1, 2, 8, 8, 8, 8, 8, 8, 2, 1, 4, 2, 8, 8, 8, 8, 8, 8, 8, 8, 1, 2, 4, 4, 2, 1, 8, 8, 8, 8, 8

Offset: 1

Amiram Eldar, Apr 15 2025

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

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.

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.

The unitary analog of A104529.
Cf. A034444, A383055 (numerators).
Similar sequences: A064608, A370898, A379514.

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

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

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

A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

Original entry on oeis.org

2, 6, 12, 3, 2, 4, 12, 12, 24, 3, 2, 4, 8, 24, 120, 15, 20, 5, 30, 20, 10, 5, 60, 30, 15, 120, 60, 15, 120, 60, 120, 40, 20, 15, 60, 120, 120, 120, 240, 240, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 1008, 1008, 1008, 1008, 63, 1008, 5040, 5040

Offset: 1

Amiram Eldar, Aug 08 2025

			Fractions begin with 1/2, 5/6, 13/12, 4/3, 3/2, 7/4, 23/12, 25/12, 53/24, 7/3, 5/2, 11/4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mikhail R. Gabdullin, Vitalii V. Iudelevich, and Sergei V. Konyagin, Karatsuba's divisor problem and related questions, arXiv:2304.04805 [math.NT], 2023.
Vitalii V. Iudelevich, On the Karatsuba divisor problem, Izvestiya: Mathematics, Vol. 86, No. 5 (2022), pp. 992-1019; arXiv preprint, arXiv:2304.03049 [math.NT], 2023.

Cf. A000005, A008329, A008864, A104529, A386921 (numerators).

Denominator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]

list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(denominator(s), ", "));}

a(n) = denominator(Sum_{k=1..n} 1/A008329(k)).

A357819 Denominators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 4, 12, 1, 8, 24, 24, 72, 72, 36, 252, 504, 126, 504, 504, 504, 2520, 840, 3360, 10080, 10080, 10080, 2016, 2016, 2016, 2016, 10080, 10080, 5040, 1260, 5040, 15120, 7560, 3780, 71820, 17955, 143640, 17955, 35910, 574560, 6320160, 6320160, 6320160, 6320160

Offset: 1

Amiram Eldar, Oct 14 2022

See A357818 for more details.

Cf. A001615, A173290, A357818 (numerators).

Similar sequences: A048049, A104529, A212718.

psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Denominator[Accumulate[1/Array[psi[#] &, 50]]]

f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = denominator(sum(k=1, n, 1/f(k))); \\ Michel Marcus, Oct 15 2022

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

cp's OEIS Frontend

A104528 Numerator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A357846 Denominators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A379358 Denominators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383056 Denominators of the partial sums of the reciprocals of the number of unitary divisors function (A034444).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A357819 Denominators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula