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.

A322744 Array T(n,k) = (3*n*k - A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 8, 8, 4, 5, 12, 11, 12, 5, 6, 14, 16, 16, 14, 6, 7, 18, 19, 24, 19, 18, 7, 8, 20, 24, 28, 28, 24, 20, 8, 9, 24, 27, 36, 33, 36, 27, 24, 9, 10, 26, 32, 40, 42, 42, 40, 32, 26, 10, 11, 30, 35, 48, 47, 54, 47, 48, 35, 30, 11, 12, 32, 40, 52, 56, 60, 60, 56, 52, 40, 32, 12
Offset: 1

Views

Author

David Lovler, Dec 24 2018

Keywords

Comments

Associative multiplication-like table whose values depend on whether n and k are odd or even.
Associativity is proved by checking the formula with eight cases of three odd and even arguments. T(n,k) is distributive as long as partitioning an even number into two odd numbers is not allowed.

Examples

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   6   8  12  14  18  20  24  26  30
   3   8  11  16  19  24  27  32  35  40
   4  12  16  24  28  36  40  48  52  60
   5  14  19  28  33  42  47  56  61  70
   6  18  24  36  42  54  60  72  78  90
   7  20  27  40  47  60  67  80  87 100
   8  24  32  48  56  72  80  96 104 120
   9  26  35  52  61  78  87 104 113 130
  10  30  40  60  70  90 100 120 130 150

Links

Crossrefs

Equals A003991 + A322630 - A319929.
Cf. A327263, A307001, A307002, A340746, A340747.
0 and diagonal is A354594.

Programs

  • Mathematica
    Table[Function[n, (3 n k - If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]])/2][m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n,k) = (3*n*k - T319929(n,k))/2;
matrix(6, 6, n, k, T(n, k)) \\ Michel Marcus, Dec 27 2018

Formula

T(n,k) = (3*n*k - (n + k - 1))/2 if n is odd and k is odd;
T(n,k) = (3*n*k - n)/2 if n is even and k is odd;
T(n,k) = (3*n*k - k)/2 if n is odd and k is even;
T(n,k) = 3*n*k/2 if n is even and k is even.
T(n,k) = 6*floor(n/2)*floor(k/2) + A319929(n,k).
T(n,n) = A354594(n). - David Lovler, Jul 09 2022

A307002 Numbers > 1 not of the form (3n*k - A319929(n,k))/2 where n and k > 1.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 10, 13, 15, 17, 21, 22, 23, 25, 29, 31, 34, 37, 39, 41, 45, 46, 49, 53, 55, 57, 58, 63, 65, 69, 71, 73, 77, 79, 81, 82, 85, 93, 94, 95, 97, 101, 105, 106, 109, 111, 118, 119, 121, 125, 129, 133, 135, 137, 141, 142, 143, 149, 151, 153, 157
Offset: 1

Views

Author

David Lovler, Mar 19 2019

Keywords

Comments

All even terms > 2 appear to be semiprimes of the form 6m+4 (A112774).
The subsequence of odd terms is A307001. - David Lovler, Jan 17 2022

Links

Crossrefs

Cf. A112774, A307001, A322630, A322744, A340746, A340747.
Third row of array A327263.

Programs

  • PARI
    T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n, k) = (3*n*k - T319929(n, k))/2; \\ A322744
lista(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, T(n, k)););); setminus([2..nn], Set(list));} \\ Michel Marcus, Jan 24 2021

Extensions

Name amended by David Lovler, Jan 25 2022

A340747 Numbers in array A322744 that do not have a unique decomposition into numbers of A307002.

Original entry on oeis.org

24, 40, 60, 67, 72, 88, 96, 100, 120, 132, 136, 144, 147, 150, 160, 168, 180, 184, 200, 204, 216, 220, 227, 232, 240, 264, 267, 276, 280, 288, 300, 307, 312, 323, 328, 330, 340, 348, 352, 360, 367, 376, 384, 387, 396, 400, 408, 420, 424
Offset: 1

Views

Author

David Lovler, Jan 20 2021

Keywords

Comments

For i >= 2, A322744(i, a(n)) is in the sequence.
There are numbers in array A322744 that have three decompositions of the form A322744(4,p) = A322744(7,q) = A322744(10,r). In these cases, p = q + r. p, q and r need not be in A307002. There are two situations. (a) For n > 0, 60n = A322744(4,10n) = A322744(7,6n) = A322744(10,4n); (b) For n >= 0, 60n+40 = A322744(4,10n+7) = A322744(7,6n+4) = A322744(10,4n+3).
A proof of p = q + r. q must be even because A322744(7,q) = even. p and r must be both odd or both even, otherwise there is the contradiction that p gets equated with a fraction. When p and r are odd, (3*4*p - 4)/2 = (3*7*q - q)/2 = (3*10*r - 10)/2. Solving for p in terms of q, and p in terms of r gives p = (5/3)*q + 1/3 and p = (5/2)*r - 1/2. Multiplying the latter by 2/3 and adding the two equations gives (5/3)*p = (5/3)*q + (5/3)*r, thus p = q + r. When p and r are even, (3*4*p)/2 = (3*7*q - q)/2 = (3*10*r)/2, and the same follows.

Examples

			60 = A322744(4,10). Also 60 = A322744(6,7) and 60 = A322744(2,20). These decompositions are the same but different from A322744(4,10) as follows. 6 = A322744(2,2) and 20 = A322744(2,7), making 60 = A322744(A322744(2,2), 7) and 60 = A322744(2, A322744(2,7)). Thus 60 can be written as A322744(2,2,7), a well-defined composition because A322744(n,k) is associative. 2,4,7 and 10 are in A307002, thus A322744(4,10) and A322744(2,2,7) are different decompositions of 60, so 60 is in the sequence.
88 is in the sequence because 88 = A322744(3,22) = A322744(4,15) and 3,4,15 and 22 are in A307002.
Examples of A322744(4,p) = A322744(7,q) = A322744(10,r) with p = q + r:
60*1 + 40 = 100 = A322744(4,17) = A322744(7,10) = A322744(10,7) and 17 = 10 + 7, which works by commuting one of the decompositions. Note that 60 also works this way. 60 = A322744(4,10) = A322744(7,6) = A322744(10,4) and 10 = 6 + 4.
60*3 = 180 = A322744(4,30) = A322744(7,18) = A322744(10,12) and 30 = 18 + 12.
60*3 + 40 = 220 = A322744(4,37) = A322744(7,22) = A322744(10,15) and 37 = 22 + 15.
See A340746 for more examples.

Links

Crossrefs

Cf. A307002, A322744, A340746.
Showing 1-3 of 3 results.

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