A104528 -id:A104528 - cp's OEIS frontend

A104529 Denominator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.

1, 2, 1, 3, 6, 12, 12, 6, 6, 12, 12, 12, 12, 6, 12, 60, 60, 20, 20, 60, 15, 60, 60, 120, 40, 40, 40, 120, 120, 30, 15, 30, 60, 15, 60, 180, 180, 45, 180, 360, 360, 45, 90, 45, 90, 180, 180, 180, 180, 180, 90, 45, 90, 360, 360, 45, 180, 90, 45, 180, 180, 90, 45, 315, 1260

Offset: 1

Emeric Deutsch, Mar 12 2005

			1,3/2,2,7/3,17/6
a(4)=3 because 1/tau(1)+1/tau(2)+1/tau(3)+1/tau(4)=1/1+1/2+1/2+1/3=7/3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A104528.

with(numtheory): a:=n->denom(sum(1/tau(k),k=1..n)): seq(a(n),n=1..70);

Denominator[Accumulate[1/#&/@DivisorSigma[0,Range[70]]]] (* Harvey P. Dale, Dec 18 2012 *)

A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

Original entry on oeis.org

1, 1, 1, 2, 7, 11, 17, 7, 3, 5, 7, 19, 25, 11, 25, 113, 143, 133, 163, 51, 14, 51, 61, 117, 391, 361, 391, 371, 431, 52, 119, 19, 81, 19, 81, 709, 799, 377, 799, 1553, 1733, 211, 467, 226, 467, 889, 979, 961, 1021, 991, 259, 503, 274, 2147, 2237, 274, 1141, 274

Offset: 1

Amiram Eldar, Oct 16 2022

			Fractions begin with 1, 1/2, 1, 2/3, 7/6, 11/12, 17/12, 7/6, 3/2, 5/4, 7/4, 19/12, ...

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

Cf. A000005, A307704, A357844 (denominators).

Similar sequences: A104528, A211177, A357820.

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

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

from fractions import Fraction
from sympy import divisor_count
def A357843(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, Oct 16 2022

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

a(n)/A357844(n) ~ n * Sum_{k=1..N} B_k/log(n)^(k-1/2) + O(n/log(n)^(N+1/2)), where B_k are constants, and in particular B_1 = (1/log(2) - 1) * (1/sqrt(Pi)) * Product_{p prime} sqrt(p^2-p) * log(p/(p-1)) (Tóth, 2017).

A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

Original entry on oeis.org

1, 2, 11, 65, 79, 6, 55, 769, 10837, 30691, 33421, 32251, 34591, 16613, 34591, 1039561, 365327, 356647, 373573, 365513, 1504367, 4400261, 4569521, 4501817, 149447, 146327, 149603, 147263, 151631, 49937, 25651, 75913, 38639, 114097, 232289, 230129, 4470731, 4408487

Offset: 1

Amiram Eldar, Oct 16 2022

			Fractions begin with 1, 2/3, 11/12, 65/84, 79/84, 6/7, 55/56, 769/840, 10837/10920, 30691/32760, 33421/32760, 32251/32760, ...

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, A065442, A065443, A068762, A357846 (denominators).

Similar sequence: A104528, A212717, A357820.

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

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

from fractions import Fraction
from sympy import divisor_sigma
def A357845(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, Oct 16 2022

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

a(n)/A357846(n) ~ E * ((2/K-1)*(log(n) + gamma + F) + 2*log(2)*K'/K^2) + O(log(n)^(5/3)*log(log(n))^(4/3)/n), where E = Product_{p prime} alpha(p), F = Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p)), alpha(p) = 1 - ((p-1)^2/p) * Sum_{k>=1} 1/((p^k-1)*(p^(k+1)-1)), beta(p) = Sum_{k>=1} k/((p^k-1)*(p^(k+1)-1)), K = A065442, K' = A065443 (Tóth, 2017).

A379357 Numerators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

Original entry on oeis.org

1, 4, 5, 11, 13, 41, 47, 122, 259, 269, 299, 152, 167, 172, 59, 4, 13, 79, 85, 43, 44, 5, 16, 161, 83, 254, 517, 29, 92, 833, 878, 6191, 6296, 6401, 6506, 26129, 27389, 27809, 28229, 5671, 5923, 5951, 6203, 6245, 6287, 6371, 6623, 33199, 33829, 34039, 34459, 34669

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 4/3, 5/3, 11/6, 13/6, 41/18, 47/18, 122/45, 259/90, 269/90, 299/90, 152/45, ...

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, A104528, A379358 (denominators).

f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[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(numerator(s), ", "))};

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

a(n)/A379358(n) = Sum_{i=1..N} b_i * n / log(n)^(i-1/3) + O(n / log(n)^(N+1-1/3)), for any fixed N >= 1, where b_i are constants. The same formula holds (with different constants) for any Piltz function d_k(n), for k >= 2, when 1/3 is replaced by 1/k.

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

Original entry on oeis.org

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

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 A104528.
Cf. A002161, A034444, A345288, A383056 (denominators).
Similar sequences: A064608, A370898, A379513.

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

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

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).

A386921 Numerators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

Original entry on oeis.org

1, 5, 13, 4, 3, 7, 23, 25, 53, 7, 5, 11, 23, 73, 377, 49, 67, 18, 113, 77, 41, 21, 257, 131, 68, 559, 287, 73, 599, 307, 629, 213, 109, 83, 337, 689, 719, 739, 1493, 1523, 4609, 4699, 33253, 34513, 34933, 35353, 36193, 36613, 37033, 37663, 38083, 7667, 7835, 7891

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, A104528, A386922 (denominators).

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

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

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

a(n)/A386922(n) <= 4 * K * n/log(n)^(3/2) + O(n*log(log(n))/log(n)^(5/2)), where K = (1/sqrt(Pi)) * Product_{p prime} sqrt(p/(p-1)) * (p * log(p/(p-1)) - 1/(p-1)) = 0.25320111501639846923... (Iudelevich, 2022). Gabdullin et al. (2023) conjectured that a(n)/A386922(n) ~ K * n/log(n)^(3/2).

A357818 Numerators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 4, 19, 7, 23, 2, 17, 53, 55, 169, 175, 89, 641, 1303, 331, 1345, 1373, 1387, 7061, 2377, 9613, 29119, 29539, 29749, 6017, 6065, 6121, 6163, 31151, 31291, 15803, 3977, 16013, 48319, 24317, 12211, 233899, 58774, 472757, 59344, 119543, 1918673, 21249043, 21336823

Offset: 1

Amiram Eldar, Oct 14 2022

			Fractions begin with 1, 4/3, 19/12, 7/4, 23/12, 2, 17/8, 53/24, 55/24, 169/72, 175/72, 89/36, ...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 100, p. 169.
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A001615, A173290, A357819 (denominators).
Cf. A001620, A065463, A335707.
Similar sequences: A028415, A104528, A212717.

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

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

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

a(n)/A357819(n) ~ C * (log(n) + gamma + D) + O(log(n)^(2/3) * log(log(n))^(4/3) / n), where C = Product_{p prime} (1 - 1/(p*(p+1))) (A065463), and D = Sum_{p prime} log(p)/(p^2+p-1) (A335707) (Sita Ramaiah and Suryanarayana, 1979; Tóth, 2017).

cp's OEIS Frontend

A104529 Denominator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.

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A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

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A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

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A379357 Numerators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

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

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A386921 Numerators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

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A357818 Numerators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

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