cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A357821 Denominators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 4, 12, 6, 24, 8, 24, 72, 72, 18, 63, 504, 63, 504, 168, 504, 2520, 2520, 10080, 1120, 3360, 3360, 672, 224, 2016, 2016, 10080, 10080, 5040, 2520, 5040, 15120, 1890, 7560, 143640, 143640, 17955, 143640, 143640, 574560, 6320160, 6320160, 6320160, 6320160
Offset: 1

Views

Author

Amiram Eldar, Oct 14 2022

Keywords

Comments

See A357820 for more details.

Crossrefs

Cf. A001615, A173290, A357820 (numerators).
Similar sequence: A211178.

Programs

  • Mathematica
    psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Denominator[Accumulate[1/Array[(-1)^(# + 1)*psi[#] &, 50]]]
  • PARI
    f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = denominator(sum(k=1, n, (-1)^(k+1)/f(k))); \\ Michel Marcus, Oct 15 2022

Formula

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

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

Views

Author

Amiram Eldar, Oct 16 2022

Keywords

Examples

			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, ...

Links

Crossrefs

Cf. A000005, A307704, A357844 (denominators).
Similar sequences: A104528, A211177, A357820.

Programs

  • Mathematica
    Numerator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]
  • PARI
    lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / numdiv(k); print1(numerator(s), ", "))};
  • Python
    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

Formula

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

Views

Author

Amiram Eldar, Oct 16 2022

Keywords

Examples

			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, ...

Links

Crossrefs

Cf. A000203, A065442, A065443, A068762, A357846 (denominators).
Similar sequence: A104528, A212717, A357820.

Programs

  • Mathematica
    Numerator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[1, #] &, 60]]]
  • PARI
    lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / sigma(k); print1(numerator(s), ", "))};
  • Python
    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

Formula

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).
Showing 1-3 of 3 results.

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