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.

A270383 Number of ordered pairs (i,j) with i >= j, |i|, |j| <= n, and |i * j| <= n.

Original entry on oeis.org

1, 6, 12, 18, 27, 33, 43, 49, 59, 68, 78, 84, 98, 104, 114, 124, 137, 143, 157, 163, 177, 187, 197, 203, 221, 230, 240, 250, 264, 270, 288, 294, 308, 318, 328, 338, 359, 365, 375, 385, 403, 409, 427, 433, 447, 461, 471, 477, 499, 508, 522
Offset: 0

Views

Author

Lorenz H. Menke, Jr., Mar 15 2016

Keywords

Examples

			For n = 2 the a(2) = 12 pairs are (2,1), (2,0), (2,-1), (1,1), (1,0), (1,-1), (1,-2), (0,0), (0,-1), (0,-2), (-1,-1), and (-1,-2). - _Danny Rorabaugh_, Apr 05 2016

Links

Crossrefs

Cf. A000005, A067274, A226355.

Programs

  • Mathematica
    a[n_]:=2Sum[Length[Divisors[k]],{k,1,n}]+Floor[Sqrt[n]]+2n+1
  • PARI
    a(n) = 2*sum(k=1, n, numdiv(k)) + sqrtint(n) + 2*n + 1; \\ Michel Marcus, Apr 05 2016

Formula

a(n) = 2*(Sum_{k=1..n} tau(k)) + floor(sqrt(n)) + 2*n + 1, where tau(k) = A000005(k) is number of divisors of k.
a(n) = A067274(n) + 2 for n >= 1.

A270543 Number of ordered pairs (i,j) of integers with |i|, |j| <= n, |i * j| <= n, and i even.

Original entry on oeis.org

1, 3, 11, 13, 25, 27, 39, 41, 57, 59, 71, 73, 93, 95, 107, 109, 129, 131, 147, 149, 169, 171, 183, 185, 213, 215, 227, 229, 249, 251, 271, 273, 297, 299, 311, 313, 341, 343, 355, 357, 385, 387, 407, 409, 429, 431, 443, 445, 481, 483, 499, 501, 521, 523, 543, 545
Offset: 0

Views

Author

Lorenz H. Menke, Jr., Mar 18 2016

Keywords

Examples

			a(0) = 1 from (i,j) = (0,0).
a(1) = 3 from (i,j) = (0,0), (0,1), (0,-1).
a(2) = 11: (0,0), (0,+-1), (0,+-2), (+-2, 0), (+-2,+-1).

Crossrefs

The corresponding sequence where i is odd is A270544.
Cf. A226355.

Programs

  • Mathematica
    a[n_]:=1+2n+2Floor[n/2]+4Sum[Floor[n/(2k)],{k,1,Floor[n/2]}]
  • PARI
    a(n) = {my(nb = 0); for (i=-n, n, if ((i % 2) == 0, for(j=-n, n, if (abs(i*j) <= n, nb++);););); nb;}

Formula

a(n) = 1 + 2*n + 2*floor(n/2) + 4*Sum_{k=1..floor(n/2)}floor(n/(2k)).
a(n) = A226355(n) - A270544(n).

Extensions

a(44) = 429 corrected by Georg Fischer, Sep 13 2023

A270544 Number of ordered pairs (i,j) with |i|, |j| <= n, |i * j| <= n, and i odd.

Original entry on oeis.org

0, 6, 10, 20, 24, 34, 42, 52, 56, 70, 78, 88, 96, 106, 114, 132, 136, 146, 158, 168, 176, 194, 202, 212, 220, 234, 242, 260, 268, 278, 294, 304, 308, 326, 334, 352, 364, 374, 382, 400, 408, 418, 434, 444, 452, 478, 486, 496, 504, 518, 530, 548, 556, 566, 582, 600
Offset: 0

Views

Author

Lorenz H. Menke, Jr., Mar 18 2016

Keywords

Examples

			a(0) = 0 from (i,j) = ().
a(1) = 6 from (i,j) = (-1,+-1), (1,+-1), (+-1,0).
a(2) = 10: (-1,+-2), (-1,+-1), (+-1,0), (1, +-2), (1,+-1).

Crossrefs

The corresponding sequence where i is even is A270543.

Programs

  • Mathematica
    a[n_] := 2 Floor[(n+1)/2] + 4 Sum[Floor[n/(2k+1)], {k,0,Floor[(n+1)/2]-1}]
  • PARI
    a(n) = {my(nb = 0); for(i=-n, n, if ((i % 2), for(j=-n, n, if (abs(i*j) <= n, nb++);););); nb;} \\ Michel Marcus, Apr 10 2016

Formula

a(n) = 2*floor((n+1)/2) + 4*Sum_{k=0..floor((n+1)/2)-1}floor(n/(2k+1)).
a(n) = A226355(n) - A270543(n).
Showing 1-3 of 3 results.

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

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