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 1-4 of 4 results.

A113415 Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 5, 1, 8, 4, 7, 3, 8, 5, 14, 1, 10, 8, 11, 4, 18, 7, 13, 3, 17, 8, 22, 5, 16, 14, 17, 1, 26, 10, 26, 8, 20, 11, 30, 4, 22, 18, 23, 7, 42, 13, 25, 3, 30, 17, 38, 8, 28, 22, 38, 5, 42, 16, 31, 14, 32, 17, 55, 1, 44, 26, 35, 10, 50, 26, 37, 8, 38, 20, 65, 11, 50, 30, 41
Offset: 1

Views

Author

Michael Somos, Oct 29 2005

Keywords

Comments

Arithmetic mean between the number of odd divisors (A001227) and their sum (A000593). This fact was essentially found by the algorithmic search of Jon Maiga's Sequence Machine, and is easily seen to be correct when compared to the PARI-program given by the original author. - Antti Karttunen, Dec 07 2021

Links

Crossrefs

Cf. A000265, A000593, A001227, A001511, A003602, A048673, A064989, A069734, A336840, A349371, A349915 (Dirichlet inverse).
Quadrisection of A349916.

Programs

  • Mathematica
    Array[DivisorSum[#, If[OddQ[#], (# + 1)/2, 0] &] &, 79] (* Michael De Vlieger, Dec 08 2021 *)
  • PARI
    a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)))

Formula

G.f.: Sum_{k>0} x^k/(1-x^(2k))^2 = Sum_{k>0} k x^(2k-1)/(1-x^(2k-1)).
a(n) = (1/2) * Sum_{d|n} (d+1)*(d mod 2). - Wesley Ivan Hurt, Nov 25 2021 [From PARI prog]
From Antti Karttunen, Dec 07 2021: (Start)
All these formulas, except the last, were found by the Sequence Machine in some form or another:
a(n) = (1/2) * (A000593(n)+A001227(n)).
a(n) = A069734(A000265(n)). [See either Rutherford's or Luschny's formula in A069734]
a(n) = A349371(n) / A001511(n).
a(n) = A349371(A000265(n)) = A336840(A064989(n)).
a(n) = a(2*n) = a(A000265(n)) = A349916(4*n).
(End)

A349915 Dirichlet inverse of A113415, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

Original entry on oeis.org

1, -1, -3, 0, -4, 3, -5, 0, 1, 4, -7, 0, -8, 5, 10, 0, -10, -1, -11, 0, 12, 7, -13, 0, -1, 8, -1, 0, -16, -10, -17, 0, 16, 10, 14, 0, -20, 11, 18, 0, -22, -12, -23, 0, -2, 13, -25, 0, -5, 1, 22, 0, -28, 1, 18, 0, 24, 16, -31, 0, -32, 17, -2, 0, 20, -16, -35, 0, 28, -14, -37, 0, -38, 20, 5, 0, 20, -18, -41, 0, -2, 22
Offset: 1

Views

Author

Antti Karttunen, Dec 07 2021

Keywords

Links

Crossrefs

Cf. A000593, A001227, A113415, A349916.

Programs

  • Mathematica
    s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)
  • PARI
    A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A113415(n/d) * a(d).
a(n) = A349916(n) - A113415(n).

A349913 Sum of A001227 (the number of odd divisors function) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 4, 0, 2, 0, 4, 8, 1, 0, 2, 0, 2, 8, 4, 0, 2, 4, 4, 4, 2, 0, 0, 0, 1, 8, 4, 8, 3, 0, 4, 8, 2, 0, 0, 0, 2, 4, 4, 0, 2, 4, 2, 8, 2, 0, 4, 8, 2, 8, 4, 0, 4, 0, 4, 4, 1, 8, 0, 0, 2, 8, 0, 0, 3, 0, 4, 4, 2, 8, 0, 0, 2, 5, 4, 0, 4, 8, 4, 8, 2, 0, 8, 8, 2, 8, 4, 8, 2, 0, 2, 4, 3, 0, 0, 0, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2021

Keywords

Links

Crossrefs

Cf. A001227 (also a quadrisection of this sequence), A327276.
Cf. also A349914, A349916.

Programs

  • Mathematica
    f1[p_,e_] := If[p==2, 1, e+1]; f2[p_, e_] := Which[e == 1, -1 - Boole[p > 2], e == 2, Boole[p > 2], e > 2, 0]; a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)
  • PARI
    A001227(n) = numdiv(n>>valuation(n, 2));
A327276(n) = sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d))); \\ From A327276
A349913(n) = (A001227(n)+A327276(n));

Formula

a(n) = A001227(n) + A327276(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A001227(d) * A327276(n/d).
a(4*n) = A001227(n).

A349914 Sum of A000593 (the sum of odd divisors function) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 8, 0, 1, 16, 12, 0, 4, 0, 16, 48, 1, 0, 10, 0, 6, 64, 24, 0, 4, 36, 28, 40, 8, 0, 0, 0, 1, 96, 36, 96, 13, 0, 40, 112, 6, 0, 0, 0, 12, 60, 48, 0, 4, 64, 26, 144, 14, 0, 40, 144, 8, 160, 60, 0, 24, 0, 64, 80, 1, 168, 0, 0, 18, 192, 0, 0, 13, 0, 76, 104, 20, 192, 0, 0, 6, 121, 84, 0, 32, 216, 88, 240
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2021

Keywords

Links

Crossrefs

Cf. A000593 (also a quadrisection of this sequence), A327278.
Cf. also A349913, A349916.

Programs

  • Mathematica
    f1[p_,e_] := If[p==2, 1, (p^(e+1)-1)/(p-1)]; f2[p_, e_] := If[p == 2, -Boole[e == 1], Which[e == 1, -p - 1, e == 2, p, e > 2, 0]]; a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)
  • PARI
    A000593(n) = sigma(n>>valuation(n, 2));
A327278(n) = sumdiv(n,d,if(d%2,d*moebius(d)*moebius(n/d),0));
A349914(n) = (A000593(n)+A327278(n));

Formula

a(n) = A000593(n) + A327278(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A000593(d) * A327278(n/d).
a(4*n) = A000593(n).
Showing 1-4 of 4 results.

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