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.

A168509 Triangle read by rows, A051731 * A101688.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Nov 27 2009

Keywords

Comments

Row sums = A079247: (1, 2, 3, 4, 4, 7, 5, 8, 8, 10,...).
A168508 = A101688 * A051731

Examples

			First few rows of the triangle =
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 0, 1, 1, 1;
1, 2, 1, 1, 1, 1,
1, 0, 0, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 0, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1;
1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1;
1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...

Crossrefs

Cf. A051731, A101688, A079247, A168508

Formula

Triangle read by rows, inverse Mobius transform of A101688; where A051731 = the
inverse Mobius transform operator.

A086670 Sum of floor(d/2) where d is a divisor of n.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 3, 7, 5, 8, 5, 13, 6, 11, 10, 15, 8, 18, 9, 20, 14, 17, 11, 29, 14, 20, 18, 27, 14, 34, 15, 31, 22, 26, 22, 44, 18, 29, 26, 44, 20, 46, 21, 41, 36, 35, 23, 61, 27, 45, 34, 48, 26, 58, 34, 59, 38, 44, 29, 82, 30, 47, 49, 63, 40, 70, 33, 62, 46, 70, 35, 96, 36, 56
Offset: 1

Views

Author

Jon Perry, Jul 27 2003

Keywords

Comments

Inverse Mobius transform of A004526. - R. J. Mathar, Jan 19 2009

Examples

			10 has divisors 1,2,5,10. floor(d/2) gives 0,1,2,5, therefore a(10)=8.

Crossrefs

Cf. A000203, A001227, A004526, A079247.
Cf. A135539.

Programs

  • Mathematica
    Table[Total[Floor[Divisors[n]/2]],{n,80}] (* Harvey P. Dale, Feb 13 2023 *)
  • PARI
    for (n=1,100,s=0; fordiv(i=n,i,s+=floor(i/2)); print1(","s))
  • PARI
    a(n) = my(f = factor(n)); (sigma(f) - (numdiv(f)/(valuation(n, 2)+1)))>>1 \\ David A. Corneth, Apr 15 2022 using Franklin T. Adams-Watters's formula

Formula

G.f.: Sum_{n>=1} floor(n/2)*x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
a(n) = (A000203(n) - A001227(n)) / 2. - Franklin T. Adams-Watters, Jan 05 2012
G.f.: Sum_{k>=1} x^(2*k) / ((1 + x^k) * (1 - x^k)^2). - Ilya Gutkovskiy, Aug 02 2021
a(n) = Sum_{i=1..floor(n/2)} A135539(n,2*i). - Ridouane Oudra, Apr 15 2022

A158951 Triangle read by rows, A051731 * A158948.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 4, 0, 1, 0, 1, 7, 2, 1, 1, 0, 1, 5, 0, 1, 0, 1, 0, 1, 8, 3, 0, 2, 0, 1, 0, 18, 0, 2, 0, 1, 0, 1, 0, 1, 10, 2, 1, 1, 1, 1, 0, 1, 0, 1
Offset: 1

Views

Author

Gary W. Adamson & Mats Granvik, Mar 31 2009

Keywords

Comments

Row sums = sigma(n), A000203. Left column = A079247

Examples

			First few rows of the triangle =
1;
2, 1;
3, 0, 1;
4, 2, 0, 1;
4, 0, 1, 0, 1;
7, 2, 1, 1, 0, 1;
5, 0, 1, 0, 1, 0, 1;
8, 3, 0, 2, 0, 1, 0, 1;
8, 0, 2, 0, 1, 0, 1, 0, 1;
10, 2, 1, 1, 1, 1, 0, 1, 0, 1;
7, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
15, 4, 1, 3, 0, 2, 0, 10, 1, 0, 1;
8, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
13, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1;
...

Crossrefs

Cf. A158948, A000203, A051731

Formula

Triangle read by rows, A051731 * A158948

A128186 A051731 * A128174.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 1, 3, 0, 2, 0, 1, 0, 1, 3, 0, 2, 0, 1, 0, 1, 0, 1, 2, 2, 1, 1, 1, 1, 0, 1, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

Left column = A001227.
Row sums = A079247: (1, 2, 3, 4, 4, 7, 5, 8, 8, 10, ...).
A128187 = A128174 * A051731.

Examples

			First few rows of the triangle:
  1;
  1, 1;
  2, 0, 1;
  1, 2, 0, 1;
  2, 0, 1, 0, 1;
  2, 2, 1, 1, 0, 1;
  2, 0, 1, 0, 1, 0, 1;
  ...

Crossrefs

Cf. A128174, A051731, A128187, A079247, A001227.

Formula

A051731 * A128174 as infinite lower triangular matrices.
Showing 1-4 of 4 results.

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