A327263 -id:A327263 - cp's OEIS frontend

A338275 Array of triples read by antidiagonals of triples, giving analogs of Pythagorean triples [a,b,c] being [odd,even,odd] consistent with the functions U(i;n,k) described in A327263.

5, 8, 9, 5, 12, 13, 15, 12, 19, 5, 16, 17, 21, 20, 29, 9, 24, 25, 5, 20, 21, 27, 28, 39, 9, 40, 41, 29, 16, 33, 5, 24, 25, 33, 36, 49, 9, 56, 57, 45, 28, 53, 23, 32, 39, 5, 28, 29, 39, 44, 59, 9, 72, 73, 61, 40, 73, 33, 56, 65, 13, 48, 49

Offset: 1

David Lovler, Oct 19 2020

In forming the triples we follow what we know about calculating Pythagorean triples given two positive integers m > n. That is, a = m^2 - n^2; b = 2m*n; c = m^2 + n^2. This is the case when i = 2. Here m is odd and n is even. The rows of triples are sorted by m then by n.
Within all rows of triples, each of a, b, and c are arithmetic progressions.
Within all rows of triples, consecutive triples have the same difference (delta) which is always an even multiple of a primitive Pythagorean triple.
  delta_a = ((m-1)^2 - n^2)/2,
  delta_b = (m-1)*n,
  delta_c = ((m-1)^2 + n^2)/2.
  When n = m - 1, delta_a = 0 while delta_b = delta_c, so delta_a^2 + delta_b^2 = delta_c^2 trivially.
In rows with n = m - 1, the following are true for all i:
  a = m + n,
  c = b + 1,
  b + c = U(i; a, a).
Within all columns of triples, for each m, b is an arithmetic progression with difference 2m+2, a has a constant second difference of -4i and c has a constant second difference of +4i.
From David Lovler, Dec 04 2020: (Start)
If we modify the Pythagorean inradius formula according to the rules of U(i;odd,even), r = (m-n)*n becomes r = (i*(m-n)*n - (i-2)*n)/2. To distinguish this from the usual inradius let us call it the inradius computation or irc. The irc might not have a Euclidean interpretation, but using it brings light to the following theorem. Within a row of the table, the inradius of the (constant, Pythagorean) difference between consecutive triples equals the difference between the ircs of consecutive triples in the row, and both equal (m-n-1)*n/2.
Proof: The left hand side, according to A338895 and A338896, equals (m-n-1)*n/2. For the right hand side, given odd m and even n, irc(i+1) - irc(i) = U(i+1; m-n, n) - U(i; m-n, n) = ((i+1)*(m-n)*n - (i+1-2)*n)/2 - (i*(m-n)*n - (i-2)*n)/2 = (m-n-1)*n/2. (End)

			In the following start of the array, the column headings would be U(i;n,k), but n and k are left out to avoid confusion with n of (m,n).
       U(1;,)       U(2;,)       U(3;,)        U(4;,)        U(5;,)       U(6;,)
m 3
n 2   [5,8,9]     [5,12,13]    [5,16,17]     [5,20,21]     [5,24,25]    [5,28,29]
-----------------------------------------------------------------------------------
m 5
n 2  [15,12,19]  [21,20,29]    [27,28,39]   [33,36,49]    [39,44,59]   [45,52,69]
n 4   [9,24,25]   [9,40,41]    [9,56,57]    [9,72,73]     [9,88,89]   [9,104,105]
-----------------------------------------------------------------------------------
m 7
n 2  [29,16,33]  [45,28,53]    [61,40,73]   [77,52,93]   [93,64,113]  [109,76,133]
n 4  [23,32,39]  [33,56,65]    [43,80,91]  [53,104,117] [63,128,143]  [73,152,169]
n 6  [13,48,49]  [13,84,85]   [13,120,121] [13,156,157] [13,192,193]  [13,228,229]
-----------------------------------------------------------------------------------
m 9
n 2  [47,20,51]  [77,36,85]   [107,52,119] [137,68,153] [167,84,187]  [197,100,221]
n 4  [41,40,57]  [65,72,97]  [89,104,137] [113,136,177] [137,168,217] [161,200,257]
n 6  [31,60,67] [45,108,117]  [59,156,167] [73,204,217] [87,252,267]  [101,300,317]
n 8  [17,80,81] [17,144,145]  [17,208,209] [17,272,273] [17,336,337]  [17,400,401]

Ron Knott, Pythagorean Triples and Online Calculators

Cf. A338895, A338896.

Given i > 0, for each (odd,even) pair (m,n) with m >= 3 and m > n >= 2, the triple [a, b, c] consists of
   a = (i*m^2 - (i-2)*(2m-1))/2 - (i*n^2)/2     analogous to m^2 - n^2
   b =  i*m*n - (i-2)*n                         analogous to 2m*n
   c = (i*m^2 - (i-2)*(2m-1))/2 + (i*n^2)/2     analogous to m^2 + n^2.

A322744 Array T(n,k) = (3nk - 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

David Lovler, Dec 24 2018

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.

			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

David Lovler, Table of n, a(n) for n = 1..861 (Antidiagonals n = 1..41, flattened)

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

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 *)

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

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

A319929 Minimal arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 0, 3, 4, 2, 2, 4, 5, 0, 5, 0, 5, 6, 2, 4, 4, 2, 6, 7, 0, 7, 0, 7, 0, 7, 8, 2, 6, 4, 4, 6, 2, 8, 9, 0, 9, 0, 9, 0, 9, 0, 9, 10, 2, 8, 4, 6, 6, 4, 8, 2, 10, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 11, 12, 2, 10, 4, 8, 6, 6, 8, 4, 10, 2, 12

Offset: 1

David Lovler, Dec 17 2018

This table is akin to multiplication in that it is associative, 1 is the identity and 0 takes any number to 0. Associativity is proved by checking eight cases of three ordered odd and even numbers. Distributivity works except if an even number is partitioned into a sum of two odd numbers.

			T(3,5) = 3 + 5 - 1 = 7, T(4,7) = 4, T(8,8) = 0.
Array T(n,k) begins:
   1  2  3  4  5  6  7  8  9 10
   2  0  2  0  2  0  2  0  2  0
   3  2  5  4  7  6  9  8 11 10
   4  0  4  0  4  0  4  0  4  0
   5  2  7  4  9  6 11  8 13 10
   6  0  6  0  6  0  6  0  6  0
   7  2  9  4 11  6 13  8 15 10
   8  0  8  0  8  0  8  0  8  0
   9  2 11  4 13  6 15  8 17 10
  10  0 10  0 10  0 10  0 10  0

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Michael De Vlieger, Array plot of T(n,k) for n = 1..150, k = 1..150 with color function indicating value, pale yellow = 0, red = 299.
David Lovler, Motivation

Cf. A322630, A322744, A327259, A327263.

Table[Function[n, If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]]][m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Mar 24 2019 *)

T(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
matrix(6, 6, n, k, T(n,k)) \\ Michel Marcus, Dec 22 2018

T(n,k) = n + k - 1 if n is odd and k is odd;
T(n,k) = n if n is even and k is odd;
T(n,k) = k if n is odd and k is even;
T(n,k) = 0 if n is even and k is even.

A327259 Array T(n,k) = 2nk - A319929(n,k), n >= 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 10, 10, 4, 5, 16, 13, 16, 5, 6, 18, 20, 20, 18, 6, 7, 24, 23, 32, 23, 24, 7, 8, 26, 30, 36, 36, 30, 26, 8, 9, 32, 33, 48, 41, 48, 33, 32, 9, 10, 34, 40, 52, 54, 54, 52, 40, 34, 10, 11, 40, 43, 64, 59, 72, 59, 64, 43, 40, 11, 12, 42, 50, 68, 72, 78, 78, 72, 68, 50, 42, 12

Offset: 1

David Lovler, Aug 27 2019

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.
T(n,k) has the same group structure as A319929, A322630 and A322744. For those arrays, position (3,3) is 5, 7 and 11 respectively. T(3,3) = 13. If we didn't have the formula for these arrays, their entries could be computed knowing one position and applying the arithmetic rules.

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   8  10  16  18  24  26  32  34  40
   3  10  13  20  23  30  33  40  43  50
   4  16  20  32  36  48  52  64  68  80
   5  18  23  36  41  54  59  72  77  90
   6  24  30  48  54  72  78  96 102 120
   7  26  33  52  59  78  85 104 111 130
   8  32  40  64  72  96 104 128 136 160
   9  34  43  68  77 102 111 136 145 170
  10  40  50  80  90 120 130 160 170 200

David Lovler, Table of n, a(n) for n = 1..465 [restored by _Georg Fischer_, Oct 14 2019]

Equals A322744 + A322630 - A319929.
Equals 4*A322630 - 3*A319929.
Cf. A327260, A327261, A327263.
0 and diagonal is A354595.

T[n_,k_]:=2n*k-If[Mod[n,2]==1,If[Mod[k,2]==1,n+k-1,k],If[Mod[k,2]==1,n,0]]; MatrixForm[Table[T[n,k],{n,1,10},{k,1,10}]] (* Stefano Spezia, Sep 05 2019 *)

T(n,k) = 2*n*k - if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
matrix(8, 8, n, k, T(n,k)) \\ Michel Marcus, Sep 04 2019

T(n,k) = 2*n*k - n - k + 1 if n is odd and k is odd;
T(n,k) = 2*n*k - n if n is even and k is odd;
T(n,k) = 2*n*k - k if n is odd and k is even;
T(n,k) = 2*n*k if n is even and k is even.
T(n,k) = 8*floor(n/2)*floor(k/2) + A319929(n,k).
T(n,n) = A354595(n). - David Lovler, Jul 09 2022
Writing T(n,k) as (4*n*k - 2*A319929(n,k))/2 shows that the array is U(4;n,k) of A327263. - David Lovler, Jan 15 2022

A322630 Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 4, 7, 4, 5, 6, 6, 8, 8, 6, 6, 7, 6, 11, 8, 11, 6, 7, 8, 8, 12, 12, 12, 12, 8, 8, 9, 8, 15, 12, 17, 12, 15, 8, 9, 10, 10, 16, 16, 18, 18, 16, 16, 10, 10, 11, 10, 19, 16, 23, 18, 23, 16, 19, 10, 11

Offset: 1

David Lovler, Dec 20 2018

This table is akin to multiplication in that it is associative, 1 is the identity and 0 takes any number to 0. Associativity is proved by checking eight cases of three ordered odd and even numbers. Distributivity works except if an even number is partitioned into a sum of two odd numbers.

Excluding the first row and the first column, every number in the table is of the form 2i*j or 2i*j - 1 where i and j > 0. Every positive even number appears in the table. Odd numbers that do not appear are of the form 2p - 1 where p is a prime number.

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   2   4   4   6   6   8   8  10  10
   3   4   7   8  11  12  15  16  19  20
   4   4   8   8  12  12  16  16  20  20
   5   6  11  12  17  18  23  24  29  30
   6   6  12  12  18  18  24  24  30  30
   7   8  15  16  23  24  31  32  39  40
   8   8  16  16  24  24  32  32  40  40
   9  10  19  20  29  30  39  40  49  50
  10  10  20  20  30  30  40  40  50  50

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened).

Cf. A076274, A327263.

0 and diagonal is A213037.

Table[Function[n, Switch[FromDigits[Mod[{n, k}, 2], 2], 0, n k/2, 1, (n k + n)/2, 2, (n k + k)/2, , (n k + n + k - 1)/2]][m - k + 1], {m, 11}, {k, m}] // Flatten (* _Michael De Vlieger, Jan 14 2022 *)

T319929(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n,k) = (T319929(n,k) + n*k)/2;
matrix(6, 6, n, k, T(n,k)) \\ Michel Marcus, Dec 22 2018

T(n,k) = (n*k + n + k - 1)/2 if n is odd and k is odd;
T(n,k) = (n*k + n)/2 if n is even and k is odd;
T(n,k) = (n*k + k)/2 if n is odd and k is even;
T(n,k) = n*k/2 if n is even and k is even.

Name clarified by David Lovler, Jan 24 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

David Lovler, Mar 19 2019

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

David Lovler, Table of n, a(n) for n = 1..4399

Cf. A112774, A307001, A322630, A322744, A340746, A340747.

Third row of array A327263.

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

Name amended by David Lovler, Jan 25 2022

A327261 Numbers > 1 not of the form 2n*k - A319929(n,k) where n and k > 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 11, 12, 14, 15, 17, 19, 21, 22, 25, 27, 28, 29, 31, 35, 37, 38, 39, 44, 45, 46, 47, 49, 51, 55, 57, 61, 62, 65, 67, 69, 71, 75, 76, 79, 81, 86, 87, 89, 91, 92, 94, 97, 99, 101, 105, 107, 109, 115, 117, 118, 119, 121

Offset: 1

David Lovler, Aug 27 2019

nonn

The subsequence of odd terms is A327260. - David Lovler, Jan 23 2022

David Lovler, Table of n, a(n) for n = 1..2858

Cf. A327259, A327260, A345357.

Fourth row of array A327263.

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n, k) = 2*n*k - T319929(n, k); \\ A327259
lista(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, T(n, k)); ); ); setminus([2..nn], Set(list)); } \\ David Lovler, Apr 30 2021

Name amended David Lovler, Jan 22 2022

A345357 Numbers m > 4 such that [m-2..m+2] belong to A327261.

Original entry on oeis.org

5, 717, 2637, 14157, 89037, 112077, 149517, 156957, 180477, 235917, 255357, 267837, 269997, 293037, 399357, 447837, 533517, 592557, 679677, 703917, 770157, 909837, 929997, 1043997, 1158237, 1257597, 1283037, 1296477, 1333197, 1369197, 1500237, 1971357, 1998717, 2062557, 2099997

Offset: 1

David Lovler, Jun 15 2021

nonn

Terms > 5 have 7 for their units digit. This is because the units digit of A327261 terms can't be 0 or 3 (row 3 of A327259 has all numbers that end in 0 or 3) and there are at most 2 consecutive even terms (see comment for A327263).

David Lovler, Table of n, a(n) for n = 1..58

Cf. A340748, A345473, A345474.

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n, k) = 2*n*k - T319929(n, k); \\ A327259
list(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, T(n, k)); ); ); setminus([1..nn], Set(list)); } \\ A327261
lista(nn) = {my(v=Vec(list(nn))); for (m=5, #v-1, my(x=v[m]); if (vecsearch(v, x-2) && vecsearch(v, x-1) && vecsearch(v, x+1) && vecsearch(v, x+2), print1(x, ", ")); ); }

A345474 Given the associative array U(n,k) described below, numbers m > 7 such that [m-5..m+5] are not in U(n,k) (excluding the first row and column).

Original entry on oeis.org

8, 106, 26874, 105834, 1080234

Offset: 1

David Lovler, Jun 22 2021

There are no more terms up to 5*10^6.
All terms equal 1 (mod 7).
U(n,k) is a commutative and associative array with integer values that depend on whether n and k are odd or even.
U(n,k) = (7*n*k - 5*(n+k-1))/2 when n and k are both odd,
U(n,k) = (7*n*k - 5*n)/2 when n is even and k is odd,
U(n,k) = (7*n*k - 5*k)/2 when n is odd and k is even and
U(n,k) = 7*n*k/2 when n and k are both even.
U(n,1) = n for all n (identity element).
U(n,0) = 0 for all n.
U(n,k) can be expressed as (7*n*k - 5*U(0;n,k))/2, where U(0;n,k) has four cases.
U(0;n,k) = n+k-1 when n and k are both odd,
U(0;n,k) = n when n is even and k is odd,
U(0;n,k) = k when n is odd and k is even and
U(0;n,k) = 0 when n and k are both even.
The ordered list of numbers > 7 that do not appear in array U(n,k) for n and k > 1 can have at most 5 consecutive even numbers and at most 7 consecutive odd numbers. See rows 2 and 3.
U(n,k) is part of a hierarchy of multiplication-like arrays, mentioned in A327263, in which the entries depend on the parity of n and k. U(0;n,k), which is A319929(n,k), is their common parity-dependent component. For i >= 0, U(i;n,k) = (i*n*k - (i-2)*U(0;n,k))/2. In the current sequence U(n,k) = U(7;n,k). Each of these arrays leaves behind a list of numbers that do not appear outside of row 1 and column 1. Think of the prime number sieve.
U(2;n,k) is normal multiplication. For i > 2, these lists are progressively more dense and include even numbers as well as odd numbers. This allows strings of consecutive integers. Here we are interested in the maximum length strings for each i.
The ordered list of numbers > i that do not appear in array U(i;n,k) for n and k > 1 can have at most i-2 consecutive even numbers and at most i consecutive odd numbers. To verify this, look at rows 2 and 3. i consecutive odd numbers cannot interleaf with i-2 even numbers, but i-1 odd numbers can. Thus the longest strings for each i are of length 2i-3. When i is even, m is odd. When i is odd, m is even.
In general, there are more terms when i is odd than when i is even. This is because there are a few ways that i consecutive odd number can overlap i-2 consecutive even numbers.
For this sequence and similar sequences constructed from U(i;n,k), all terms m == 1 (mod i). To prove this, look at gaps of 2i-3 in row 2 of U(i;n,k). The longest strings of consecutive numbers not in U(i;n,k) can occur only for these 2i-3 numbers. The second number before any of the gaps is an even number of the form U(i;2,e) = i*2*e/2 == 0 (mod i) (where e is an even number). The middle of the string, m = U(i;2,e) + i + 1. Thus m == 1 (mod i).
Likewise, for any i, all terms m == 2 (mod i+1).
The following observations are included to expand upon the theme. For odd i in [9..23], the number of terms in [i+2..5*10^6] represented as [i, number of terms] are [9, 3], [11, 8], [13, 3], [15, 0], [17, 3], [19, 3], [21, 0], [23, 3]. For odd i in [25..99], 11 i's have no terms in [i+2..5*10^5], 12 have 1 such term, 9 have 2, 4 have 3, 1 has 4 and 1 has 5 such terms.
The scarcity of terms for even i's is borne out by the observation that up to 5*10^6, for even i in [6..22], the only terms > i+1 occur when i = 14 (1667), i = 20 (3341 and 1663181) and i = 22 (16171). Continuing the observation for even i in [24..140], up to 5*10^5 only 14 i's have a term > i+1.

			Array U(0;n,k) = A319929(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   0   2   0   2   0   2   0   2   0
   3   2   5   4   7   6   9   8  11  10
   4   0   4   0   4   0   4   0   4   0
   5   2   7   4   9   6  11   8  13  10
   6   0   6   0   6   0   6   0   6   0
   7   2   9   4  11   6  13   8  15  10
   8   0   8   0   8   0   8   0   8   0
   9   2  11   4  13   6  15   8  17  10
  10   0  10   0  10   0  10   0  10   0
Array U(n,k) = U(7;n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2  14  16  28  30  42  44  56  58  70
   3  16  19  32  35  48  51  64  67  80
   4  28  32  56  60  84  88 112 116 140
   5  30  35  60  65  90  95 120 125 150
   6  42  48  84  90 126 132 168 174 210
   7  44  51  88  95 132 139 176 183 220
   8  56  64 112 120 168 176 224 232 280
   9  58  67 116 125 174 183 232 241 290
  10  70  80 140 150 210 220 280 290 350
Numbers up to 200 not in U(n,k) (excluding row 1 and column 1): 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 33, 34, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 59, 61, 62, 63, 66, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 85, 87, 89, 91, 92, 93, 94, 97, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 117, 118, 119, 121, 122, 123, 124, 127, 129, 130, 133, 134, 135, 136, 137, 138, 141, 143, 145, 146, 148, 149, 151, 152, 153, 157, 158, 159, 161, 162, 164, 165, 166, 167, 169, 171, 173, 175, 177, 178, 181, 186, 187, 188, 189, 190, 191, 193, 194, 197, 199.

David Lovler, Terms for i <= 140

In A327263 U(n,k) is called U(7;n,k).

Cf. A340748, A345357, A345473.

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
U(n, k) = (7*n*k - 5*T319929(n, k))/2;
list(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, U(n, k)); ); ); setminus([1..nn], Set(list)); }
lista(nn) = {my(v=Vec(list(nn))); for (m=8, #v-1, my(x=v[m]); if (#setintersect(v, [x-5..x+5])==11, print1(x, ", ")); ); }

/* This program computes terms of sequences based on U(i;n,k) for i >= 2. */
T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
U(n, k) = (i*n*k - (i-2)*T319929(n, k))/2; \\ U(i; n, k)
list(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, U(n, k)); ); ); setminus([1..nn], Set(list)); }
lista(nn) = {my(v=Vec(list(nn))); for (m=i+1, #v-1, my(x=if(Mod(v[m],i)==1,v[m])); if (#setintersect(v,[x-i+2..x+i-2])==2*i-3, print1(x, ", ")); ); }
/* Type for example: i=4; lista(10^6) */

A345473 Given the associative array U(n,k) described below, numbers m > 5 such that [m-3..m+3] are not in U(n,k) (excluding the first row and column).

Original entry on oeis.org

6, 56, 236, 956, 2636, 3356, 6236, 9716, 10196, 13436, 15896, 18296, 24716, 26396, 36116, 36956, 37196, 42956, 53036, 69356, 82556, 84536, 119516, 121496, 181556, 201116, 204236, 221756, 252116, 259676, 332636, 359036, 365036, 401516

Offset: 1

David Lovler, Jun 21 2021

nonn

U(n,k) is a commutative and associative array with integer values that depend on whether n and k are odd or even.
U(n,k) = (5*n*k - 3*(n+k-1))/2 when n and k are both odd.
U(n,k) = (5*n*k - 3*n)/2 when n is even and k is odd.
U(n,k) = (5*n*k - 3*k)/2 when n is odd and k is even.
U(n,k) = 5*n*k/2 when n and k are both even.
U(n,1) = n for all n (identity element).
U(n,0) = 0 for all n.
The ordered list of numbers >5 that do not appear in array U(n,k) for n and k > 1 can have at most 3 consecutive even numbers and at most 5 consecutive odd numbers. See rows 2 and 3.
The terms all end in 6 because row 2 of U(n,k) has all numbers that end in 0 or 2 and there are at most 3 consecutive even numbers in the set of numbers not in array U(n,k) excluding the first row and column (see comment for A327263).
There are 119 terms up to 5*10^6.

			Array U(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2  10  12  20  22  30  32  40  42  50
   3  12  15  24  27  36  39  48  51  60
   4  20  24  40  44  60  64  80  84 100
   5  22  27  44  49  66  71  88  93 110
   6  30  36  60  66  90  96 120 126 150
   7  32  39  64  71  96 103 128 135 160
   8  40  48  80  88 120 128 160 168 200
   9  42  51  84  93 126 135 168 177 210
  10  50  60 100 110 150 160 200 210 250
Numbers up to 100 not in U(n,k) (excluding row 1 and column 1): 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 21, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 41, 43, 45, 46, 47, 53, 54, 55, 56, 57, 58, 59, 61, 65, 67, 68, 69, 73, 74, 76, 77, 78, 79, 81, 83, 85, 86, 89, 91, 94, 95, 97, 98.

David Lovler, Table of n, a(n) for n = 1..119

In A327263 U(n,k) is called U(5;n,k).

Cf. A340748, A345357, A345474.

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
U(n, k) = (5*n*k - 3*T319929(n, k))/2;
list(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, U(n, k)); ); ); setminus([1..nn], Set(list)); }
lista(nn) = {my(v=Vec(list(nn))); for (m=6, #v-1, my(x=v[m]); if (#setintersect(v,[x-3..x+3])==7, print1(x, ", ")); ); }

cp's OEIS Frontend

A338275 Array of triples read by antidiagonals of triples, giving analogs of Pythagorean triples [a,b,c] being [odd,even,odd] consistent with the functions U(i;n,k) described in A327263.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A319929 Minimal arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A327259 Array T(n,k) = 2*n*k - A319929(n,k), n >= 1, k >= 1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322630 Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A327261 Numbers > 1 not of the form 2n*k - A319929(n,k) where n and k > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A345357 Numbers m > 4 such that [m-2..m+2] belong to A327261.

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

A327259 Array T(n,k) = 2nk - A319929(n,k), n >= 1, k >= 1, read by antidiagonals.