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.

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A339026 Number of pairs (x,y): 1 <= x < y <= n*n, such that x*y is a square.

Original entry on oeis.org

0, 1, 4, 8, 16, 27, 40, 58, 80, 105, 128, 158, 198, 237, 278, 336, 382, 435, 502, 574, 632, 699, 798, 868, 968, 1063, 1176, 1262, 1362, 1505, 1590, 1732, 1880, 2007, 2150, 2272, 2434, 2567, 2746, 2930, 3062, 3265, 3444, 3654, 3832, 4035, 4284, 4454, 4668, 4875, 5142, 5364, 5590, 5889, 6078, 6336, 6614, 6881, 7166
Offset: 1

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Author

Edward Krogius, Nov 19 2020

Keywords

Examples

			For n = 3, we have the following solutions: (1,4), (1,9), (2,8), (4,9), therefore a(3) = 4.
For n = 4, we have the following solutions: (1,4), (1,9), (1,16), (2,8), (3,12), (4,9), (4,16), (9,16), therefore a(4) = 8.

Links

Crossrefs

Cf. A000010, A015614, A018805, A132188, A132189, A132345, A338894.

Programs

  • Mathematica
    Array[Sum[EulerPhi[j] Floor[(#^2)/(j^2)], {j, 2, #}] &, 59] (* Michael De Vlieger, Dec 11 2020 *)
  • PARI
    A339026(n) = sum(i=2,n,floor(n^2/i^2)*eulerphi(i)); \\ Antti Karttunen, Nov 23 2020

Formula

a(n) = Sum_{j=2..n} phi(j) * floor(n^2/j^2).
a(n) = (A338894(n) - n^2)/2.
a(n) = A132189(n^2)/2. - Antti Karttunen, Nov 23 2020

A350365 Array read by antidiagonals: T(n,k) is the number of sequences of length 2*n+1 with terms in 0..k such that the Hankel matrix of the sequence is singular, but the Hankel matrix of any proper subsequence with an odd number of consecutive terms is invertible, n, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 6, 6, 10, 0, 0, 1, 7, 16, 52, 0, 0, 0, 1, 8, 36, 148, 116, 8, 0, 0, 1, 9, 58, 448, 644, 528, 12, 0, 0, 1, 12, 82, 885, 2932, 4032, 1326, 0, 0, 0
Offset: 0

Views

Author

Pontus von Brömssen, Dec 27 2021

Keywords

Comments

T(n,2) = 0 for n = 4 and for n >= 7.

Examples

			Array begins:
  n\k|  0  1  2   3   4    5
  ---+----------------------
   0 |  1  1  1   1   1    1
   1 |  0  1  2   3   6    7
   2 |  0  0  2   6  16   36
   3 |  0  0 10  52 148  448
   4 |  0  0  0 116 644 2932
For n = 2 and k = 4, the following T(2,4) = 16 sequences are counted:
  (1, 1, 2, 2, 4),
  (1, 2, 1, 2, 1),
  (1, 2, 2, 4, 4),
  (1, 3, 1, 3, 1),
  (1, 4, 1, 4, 1),
  (2, 1, 2, 1, 2),
  (2, 3, 2, 3, 2),
  (2, 4, 2, 4, 2),
  (3, 1, 3, 1, 3),
  (3, 2, 3, 2, 3),
  (3, 4, 3, 4, 3),
  (4, 1, 4, 1, 4),
  (4, 2, 2, 1, 1),
  (4, 2, 4, 2, 4),
  (4, 3, 4, 3, 4),
  (4, 4, 2, 2, 1).

Crossrefs

Cf. A350330, A350364.
Cf. A000012 (row n = 0), A132188 (row n = 1), A000007 (column k = 0), A019590 (column k = 1).

A362936 a(n) = A000982(n) - A362931(n).

Original entry on oeis.org

0, 0, 2, 4, 8, 8, 14, 20, 28, 36, 46, 52, 64, 76, 86, 100, 116, 128, 146, 160, 180, 200, 222, 240, 264, 288, 314, 336, 364, 384, 414, 444, 476, 508, 538, 568, 604, 640, 678, 712, 752, 784, 826, 868, 904, 948, 994, 1036, 1084, 1132, 1182, 1232, 1284, 1332, 1386, 1436, 1492, 1548, 1606, 1652, 1712
Offset: 1

Views

Author

N. J. A. Sloane, Aug 28 2023

Keywords

Comments

a(n) = (number of pairs (i,j) in [1..n] X [1..n] with integral arithmetic mean) - (number of pairs (i,j) in [1..n] X [1..n] with integral harmonic mean).

Links

Crossrefs

Cf. A000982, A132188, A362931-A932937.

Programs

  • Python
    def A362936(n): return ((n-1)**2>>1)-(sum(1 for x in range(1,n+1) for y in range(1,x) if not (x*y<<1)%(x+y))<<1) # Chai Wah Wu, Aug 29 2023
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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