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.

A088839 Numerator of sigma(4n)/sigma(n).

Original entry on oeis.org

7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 85, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 511, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31, 7, 5, 7, 127, 7, 5, 7, 31, 7, 5, 7, 21, 7, 5, 7, 31
Offset: 1

Views

Author

Labos Elemer, Nov 04 2003

Keywords

Crossrefs

Programs

  • Maple
    f:= proc(n) local m;
      m:= padic:-ordp(n,2);
      if m::odd then (2^(m+3)-1)/3 else 2^(m+3)-1 fi
    end proc:
    map(f, [$1..200]); # Robert Israel, Nov 19 2017
  • Mathematica
    k=4; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
  • PARI
    A088839(n) = numerator(sigma(4*n)/sigma(n)); \\ Antti Karttunen, Nov 18 2017

Formula

a(n) = (8*A006519(n)-1)/(1+2*A096268(n)). - Robert Israel, Nov 19 2017
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A193553(n)/A000203(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088840(k) = 3*A065442 + 1 = 5.820085... . (End)

Extensions

Typo in definition corrected by Antti Karttunen, Nov 18 2017

A088840 Denominator of sigma(4n)/sigma(n).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 127, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1
Offset: 1

Views

Author

Labos Elemer, Nov 04 2003

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Denominator[DivisorSigma[1, 4*n]/DivisorSigma[1, n]], {n, 1, 128}]
    a[n_] := Module[{e = IntegerExponent[n, 2]}, (((-1)^e+2)*(2^(e+1)-1))/3]; Array[a, 100] (* Amiram Eldar, Oct 03 2023 *)
  • PARI
    A088840(n) = denominator(sigma(4*n)/sigma(n)); \\ Antti Karttunen, Nov 18 2017
    
  • PARI
    a(n) = {my(e = valuation(n, 2)); (((-1)^e+2) * (2^(e+1)-1))/3;} \\ Amiram Eldar, Oct 03 2023

Formula

From Amiram Eldar, Oct 03 2023: (Start)
Multiplicative with a(2^e) = (((-1)^e+2)*(2^(e+1)-1))/3 = A213243(e+1), and a(p^e) = 1 for an odd prime p.
a(n) = A213243(A007814(n+1)).
Dirichlet g.f.: ((8^s + 4^s + 2^(s+1))/(8^s + 4^s - 2^(s+2) - 4)) * zeta(s).
Sum_{k=1..n} a(k) = (2*n/(3*log(2))) * (log(n) + gamma - 1 + 7*log(2)/12), where gamma is Euler's constant (A001620). (End)

Extensions

Typo in definition corrected by Antti Karttunen, Nov 18 2017

A088842 Denominator of the quotient sigma(7n)/sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8
Offset: 1

Views

Author

Labos Elemer, Nov 04 2003

Keywords

Comments

Sum of powers of 7 dividing n. - Amiram Eldar, Nov 27 2022

Crossrefs

Cf. A000203 (sigma), A001620, A088841 (numerators), A283078 (sigma(7n)).

Programs

  • Mathematica
    Table[Denominator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}] (* corrected by Ilya Gutkovskiy, Dec 15 2020 *)
    a[n_] := (7^(IntegerExponent[n, 7] + 1) - 1)/6; Array[a, 100] (* Amiram Eldar, Nov 27 2022 *)
  • PARI
    a(n) = denominator(sigma(7*n)/sigma(n)); \\ Michel Marcus, Dec 15 2020
    
  • PARI
    a(n) = (7^(valuation(n, 7) + 1) - 1)/6; \\ Amiram Eldar, Nov 27 2022

Formula

G.f.: Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k)). - Ilya Gutkovskiy, Dec 15 2020
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = 1 for p != 7.
Dirichlet g.f.: zeta(s) / (1 - 7^(1 - s)).
Sum_{k=1..n} a(k) ~ n*log_7(n) + (1/2 + (gamma - 1)/log(7))*n, where gamma is Euler's constant (A001620). (End)
Showing 1-3 of 3 results.