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.

A116073 Sum of the divisors of n that are not divisible by 5.

Original entry on oeis.org

1, 3, 4, 7, 1, 12, 8, 15, 13, 3, 12, 28, 14, 24, 4, 31, 18, 39, 20, 7, 32, 36, 24, 60, 1, 42, 40, 56, 30, 12, 32, 63, 48, 54, 8, 91, 38, 60, 56, 15, 42, 96, 44, 84, 13, 72, 48, 124, 57, 3, 72, 98, 54, 120, 12, 120, 80, 90, 60, 28, 62, 96, 104, 127, 14, 144, 68, 126, 96, 24
Offset: 1

Views

Author

Michael Somos, Feb 04 2006

Keywords

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 463 Entry 4(i).

Links

Crossrefs

Cf. A028887(n) = 6*a(n) if n>0.
Cf. A145466.
Cf. A091703, A035207 (number of divisors of n that are not divisible by 5).

Programs

  • Mathematica
    f[p_, e_] := If[p == 5, 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)
  • PARI
    a(n) = if(n<1, 0, sumdiv(n,d,(d%5>0)*d))

Formula

a(n) is multiplicative with a(5^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) otherwise.
G.f.: Sum_{k>0} k*x^k/(1-x^k) - 5*k*x^(5*k)/(1-x^(5*k)).
L.g.f.: log(Product_{k>=1} (1 - x^(5*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
Sum_{k=1..n} a(k) ~ (Pi^2/15) * n^2. - Amiram Eldar, Oct 04 2022
Inverse Mobius transf. of A091703. Dirichlet g.f. (1-5^(1-s))*zeta(s-1)*zeta(s). - R. J. Mathar, May 17 2023

A035191 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 9.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 2, 4, 1, 4, 2, 3, 2, 4, 2, 5, 2, 2, 2, 6, 2, 4, 2, 4, 3, 4, 1, 6, 2, 4, 2, 6, 2, 4, 4, 3, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 5, 3, 6, 2, 6, 2, 2, 4, 8, 2, 4, 2, 6, 2, 4, 2, 7, 4, 4, 2, 6, 2, 8, 2, 4, 2, 4, 3, 6, 4, 4, 2, 10, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of divisors of n not congruent to 0 mod 3. - Vladeta Jovovic, Oct 26 2001
a(n) is the number of factors (over Q) of the polynomial x^(2n) + x^n + 1 . a(n) = d(3n) - d(n) where d() is the divisor function. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Equals Mobius transform of A011655. - Gary W. Adamson, Apr 24 2009

Links

Crossrefs

Cf. A011655, A035207, A046913, A054584.
Cf. A000005, A001227, A001620, A001817, A001817, A038502.

Programs

  • Haskell
    a035191 n = a001817 n + a001822 n  -- Reinhard Zumkeller, Nov 26 2011
  • Magma
    [NumberOfDivisors(n)/Valuation(3*n, 3): n in [1..100]]; // Vincenzo Librandi, Jun 03 2019
  • Maple
    for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=3 then b := 1:else b := e+1:fi:s := s*b:od:printf(`%d,`,s); od:
# alternative
A035191 := proc(n)
    A001817(n)+A001822(n) ;
end proc:
[seq(A035191(n),n=1..100)] ; # R. J. Mathar, Sep 25 2017
  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[9, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
f[3, e_] := 1; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)
  • PARI
    my(m=9); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(9, d)); \\ Amiram Eldar, Nov 20 2023

Formula

Multiplicative with a(3^e)=1 and a(p^e)=e+1 for p<>3.
G.f.: Sum_{k>0} x^k*(1+x^k)/(1-x^(3*k)). - Vladeta Jovovic, Dec 16 2002
a(n) = A001817(n) + A001822(n). [Reinhard Zumkeller, Nov 26 2011]
a(n) = tau(3*n) - tau(n). - Ridouane Oudra, Sep 05 2020
From Amiram Eldar, Nov 27 2022: (Start)
Dirichlet g.f.: zeta(s)^2 * (1 - 1/3^s).
Sum_{k=1..n} a(k) ~ (2*n*log(n) + (4*gamma + log(3) - 2)*n)/3, where gamma is Euler's constant (A001620). (End)
a(n) = Sum_{d|n} Kronecker(9, d). - Amiram Eldar, Nov 20 2023
a(n) = A000005(A038502(n)). - Ridouane Oudra, Sep 30 2024

A327785 Square array read by antidiagonals: A(n,k) = Sum_{d|n} (k/d), (n>=1, k>=0), where (m/n) is the Kronecker symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 0, 3, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 0, 4, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 2, 1, 1, 2, 0, 2, 4, 1, 1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 1, 2, 0, 3, 2, 0, 2, 0, 1, 4, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 3, 0, 4, 0, 0, 3, 0, 0, 6, 1
Offset: 1

Views

Author

Seiichi Manyama, Sep 25 2019

Keywords

Examples

			Square array begins:
   1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 2, 1, 0, 1, 0, 1, 2, ...
   1, 2, 0, 1, 2, 0, 1, 2, ...
   1, 3, 1, 1, 1, 1, 1, 3, ...
   1, 2, 0, 0, 2, 1, 2, 0, ...
   1, 4, 0, 0, 2, 0, 1, 4, ...
   1, 2, 2, 0, 2, 0, 0, 1, ...
   1, 4, 1, 0, 1, 0, 1, 4, ...

Links

Crossrefs

Columns k=0..31 give A000012, A000005, A035185, A035186, A001227, A035187, A035188, A035189, A035185, A035191, A035192, A035193, A035194, A035195, A035196, A035197, A001227, A035199, A035200, A035201, A035202, A035203, A035204, A035205, A035188, A035207, A035208, A035186, A035210, A035211, A035212, A035213.
Cf. A215200.

Programs

  • Mathematica
    A[n_, k_] := Sum[KroneckerSymbol[k, d], {d, Divisors[n]}];
Table[A[n - k, k], {n, 1, 13}, {k, n - 1, 0, -1}] // Flatten (* Jean-François Alcover, Sep 25 2019 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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