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 1-7 of 7 results.

A080278 a(n) = (3^(v_3(n) + 1) - 1)/2, where v_3(n) = highest power of 3 dividing n = A007949(n).

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 40, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 40, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 121, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Mar 19 2003

Keywords

Comments

Denominator of the quotient sigma(3*n)/sigma(n). - Labos Elemer, Nov 04 2003
a(n) = b/(3*(c+d)) where b, c, d are the sums of the divisors of 3*n that are congruent respectively to 0, 1 and 2 mod 3. - Michel Lagneau, Nov 05 2012
Sum of powers of 3 dividing n. - Amiram Eldar, Nov 27 2022

Examples

			a(6) = 4 because the divisors of 3*6 = 18 are {1, 2, 3, 6, 9, 18} => b = 3 + 6 + 9 + 18 = 36, c = 1, d = 2, hence a(6) = b/(3*(c+d)) = 36/(3*(1+2)) = 36/9 = 4. - _Michel Lagneau_, Nov 05 2012

Links

Crossrefs

Cf. A000203, A001620, A007949, A080333, A088838 (numerator of sigma(3*n)/sigma(n)).

Programs

  • Maple
    A080278 := n->(3^(A007949(n)+1)-1)/2;
  • Mathematica
    Table[Denominator[DivisorSigma[1, 3*n]/DivisorSigma[1, n]], {n, 1, 128}]
a[n_] := (3^(IntegerExponent[n, 3] + 1) - 1)/2; Array[a, 100] (* Amiram Eldar, Nov 27 2022 *)
  • PARI
    a(n) = denominator(sigma(3*n)/sigma(n)); \\ Michel Marcus, Dec 15 2019
  • PARI
    a(n) = (3^(valuation(n, 3) + 1) - 1)/2; \\ Amiram Eldar, Nov 27 2022

Formula

G.f.: Sum_{k>=0} 3^k*x^(3^k)/(1-x^(3^k)). - Ralf Stephan, Jun 15 2003
L.g.f.: -log(Product_{k>=0} (1 - x^(3^k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 15 2018
a(n) = sigma(n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = 1 for p != 3.
Dirichlet g.f.: zeta(s) / (1 - 3^(1 - s)).
Sum_{k=1..n} a(k) ~ n*log_3(n) + (1/2 + (gamma - 1)/log(3))*n, where gamma is Euler's constant (A001620). (End)

A088837 Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.

Original entry on oeis.org

3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 127, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 255, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3, 7, 3, 63, 3, 7, 3, 15, 3, 7, 3, 31, 3, 7, 3, 15, 3
Offset: 1

Views

Author

Labos Elemer, Nov 04 2003

Keywords

Comments

In general sigma(2^k*n) / sigma(n) = ((2^k*n) XOR (2^k*n-1)) / (n XOR (n-1)), see link. Jon Maiga, Dec 10 2018

Links

Crossrefs

Cf. A000203, A038712 (denominators), A062731, A088838, A088839, A088840, A080278, A220466.
Cf. A001620, A065442.

Programs

  • Maple
    nmax:=93: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2^(p+2)-1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 09 2013
  • Mathematica
    k=2; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
Table[BitXor[2*n, 2*n - 1], {n, 128}] (* Jon Maiga, Dec 10 2018 *)
  • PARI
    A088837(n) = numerator(sigma(n<<1)/sigma(n)); \\ Antti Karttunen, Nov 01 2018

Formula

a(n) = 4*2^A007814(n)-1 = 4*A006519(n)-1 = A059159(n)-1 = 2*A038712(n) + 1.
a((2*n-1)*2^p) = 2^(p+2)-1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 09 2013
a(n) = (2n) XOR (2n-1). - Jon Maiga, Dec 10 2018
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A062731(n)/A000203(n)).
Sum_{k=1..n} a(k) ~ (log_2(n) + (gamma-1)/log(2) + 1)*2*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A038712(k) = A065442 + 1 = 2.606695... . (End).

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

Links

Crossrefs

For denominator see A088840.
Cf. A000203, A038712, A088837, A088838, A088841, A080278, A193553.
Cf. A006519, A065442, A096268.

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

Links

Crossrefs

See A088839 for numerator.
Cf. A088837, A088838, A088841, A038712, A080278, A000203, A001620, A007814, A193553, A213243.

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

Links

Crossrefs

Cf. A000203 (sigma), A001620, A088841 (numerators), A283078 (sigma(7n)).
Cf. A080278, A038712, A088837, A088838, A088839, A088840.

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)

A088841 Numerator of the quotient sigma(7*n)/sigma(n).

Original entry on oeis.org

8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 400, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8
Offset: 1

Views

Author

Labos Elemer, Nov 04 2003

Keywords

Links

Crossrefs

Cf. A088837, A088838, A088839, A088840, A088841, A088842 (denominators).
Cf. A000203, A038712, A080278, A214411, A268354, A283078.

Programs

  • Mathematica
    Table[Numerator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}]
  • PARI
    a(n) = numerator(sigma(7*n)/sigma(n)); \\ Amiram Eldar, Mar 22 2024

Formula

From Amiram Eldar, Mar 22 2024: (Start)
a(n) = numerator(A283078(n)/A000203(n)).
a(n) = (7^(A214411(n)+2)-1)/6 = (49*A268354(n)-1)/6.
Sum_{k=1..n} a(k) ~ (7/log(7))*n*log(n) + (9/2 + 7*(gamma-1)/log(7))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088842(k) = 1 + 36 * Sum_{k>=1} 1/(7^k-1) = 7.87276224676... . (End)

A074724 Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 243, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3
Offset: 1

Views

Author

Benoit Cloitre, Sep 04 2002

Keywords

Comments

If m == 1, 2 or 3 (mod 4) then F(m) is not divisible by 3.

Links

Crossrefs

Cf. A000045, A000203, A038500, A033888, A051064, A074724, A088838.

Programs

  • Mathematica
    Table[3^IntegerExponent[Fibonacci[4n],3],{n,100}] (* Harvey P. Dale, Jun 03 2012 *)
  • PARI
    a(n) = 3^valuation(fibonacci(4*n), 3); \\ Michel Marcus, May 13 2022

Formula

If k == 1 or 2 (mod 3) then a(3^m*k) = 3^(m+1) for m>=0.
a(n) = A038500(A033888(n)). - Amiram Eldar, May 13 2022
a(n) = 3^A051064(n) (conjectured). - Michel Marcus, May 17 2022
Conjecture: a(n) = (sigma(3*n) - sigma(n))/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). Equivalently, a(n) = A088838(n) - A074724(n). - Peter Bala, Jun 10 2022
Showing 1-7 of 7 results.

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