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.

A357820 Numerators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 2, 11, 3, 11, 5, 23, 7, 23, 65, 71, 17, 64, 491, 64, 491, 173, 505, 2651, 2581, 10639, 1151, 3593, 3523, 727, 237, 2189, 2147, 11071, 10931, 5623, 2759, 5623, 16589, 2113, 8347, 162373, 159979, 20318, 160549, 163969, 649891, 7292441, 7204661, 7292441, 7204661
Offset: 1

Views

Author

Amiram Eldar, Oct 14 2022

Keywords

Examples

			Fractions begin with 1, 2/3, 11/12, 3/4, 11/12, 5/6, 23/24, 7/8, 23/24, 65/72, 71/72, 17/18, ...

Links

Crossrefs

Cf. A001615, A173290, A357821 (denominators).
Cf. A001620, A065463, A335707.
Similar sequence: A211177.

Programs

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

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/psi(k)).
a(n)/A357821(n) ~ (C/5) * (log(n) + gamma + D + 24*log(2)/5) + 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) (Bordellès and Cloitre, 2013; Tóth, 2017).

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

Views

Author

Amiram Eldar, Oct 16 2022

Keywords

Comments

See A357843 for more details.

Links

Crossrefs

Cf. A000005, A307704, A357843 (numerators).
Similar sequences: A104529, A211178, A357821.

Programs

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

Formula

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

Views

Author

Amiram Eldar, Oct 16 2022

Keywords

Comments

See A357845 for more details.

Links

Crossrefs

Cf. A000203, A068762, A357845 (numerators).
Similar sequence: A104529, A212718, A357821.

Programs

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

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/sigma(k)), where sigma(k) = A000203(k).
Showing 1-3 of 3 results.

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