A340747 -id:A340747 - cp's OEIS frontend

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

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

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

A340746 Numbers in array A322744 that do not have a unique decomposition into numbers of A307002 and are not equal to A322744(n,k), n > 1, k in the sequence.

Original entry on oeis.org

24, 40, 60, 67, 88, 100, 132, 136, 147, 150, 184, 204, 220, 227, 232, 276, 307, 323, 328, 330, 340, 348, 367, 376, 387, 424, 460, 472, 484, 492, 499, 510, 547, 550, 564, 567, 568, 580, 627, 636, 664, 675, 690, 707, 708, 712, 726, 748, 767

Offset: 1

David Lovler, Jan 18 2021

nonn

While array A322744 has many properties of the multiplication table, one way the numbers that sieve out of the array fail to be prime numbers is that unique factorization does not hold. Some numbers have two or more decompositions.

The numbers in this sequence are primitive in the sense that they are not A322744 multiples of an earlier number in the sequence.

			24 = A322744(4,4). Also 24 = A322744(6,3) and 24 = A322744(2,8). These two decompositions are the same but they differ from A322744(4,4) as follows. 6 = A322744(2,2) and 8 = A322744(2,3), making 24 = A322744(A322744(2,2), 3) and 24 = A322744(2, A322744(2,3)). Thus 24 can be written as A322744(2,2,3), a well-defined composition because A322744(n,k) is associative. 2,3 and 4 are in A307002, thus A322744(4,4) and A322744(2,2,3) are different decompositions of 24, so 24 is in the sequence.
40 is in the sequence because 40 = A322744(3,10) = A322744(4,7) and 3,4,7 and 10 are in A307002.
67 is in the sequence because 67 = A322744(3,17) = A322744(7,7) and 3,7 and 17 are in A307002.
220 has three decompositions. 220 = A322744(4,37) = A322744(7,22) = A322744(10,15) and 4,7,10,15,22 and 37 are in A307002.
72 = A322744(2,2,2,3) = A322744(2,4,4) is not in the sequence because 72 = A322744(2,24) and 24 is in the sequence.

Cf. A307002, A322744, A340747.

T[n_,k_]:=T[n,k]=(3*n*k-If[OddQ@n,If[OddQ@k,n+k-1,k],If[OddQ@k,n,0]])/2;F[d_]:=If[(q=Union[Sort/@(Position[Table[T[n,k],{n,2,Ceiling[d/3]},{k,2,Ceiling[d/3]}],d]+1)])=={},{{d}},q];FC[x_]:=FixedPoint[Union[Sort/@Flatten[Flatten/@Tuples[#]&/@((F/@#&/@#)&[#]),1]]&,F[x]];list={};Do[If[Length@FC@i>1&&ContainsNone[list,Flatten@F@i],AppendTo[list,i]],{i,500}];list (* Giorgos Kalogeropoulos, Apr 11 2021 *)

A348824 Numbers in array A327259 that do not have a unique decomposition into numbers of A327261.

Original entry on oeis.org

32, 48, 72, 96, 112, 126, 128, 144, 160, 168, 176, 192, 198, 221, 224, 240, 252, 256, 264, 288, 294, 304, 336, 342, 347, 352, 360, 368, 384, 392, 396, 414, 416, 432, 448, 456, 462, 480, 496, 504, 512, 528, 544, 545, 552, 558, 560, 576, 588, 599

Offset: 1

David Lovler, Oct 31 2021

nonn

While array A327259 has many properties of the multiplication table, one way the numbers that sieve out of the array fail to be prime numbers is that unique factorization does not hold. Some numbers have two or more decompositions.

For i >= 2, A327259(i, a(n)) is in the sequence.

			48 is in the sequence because 48 = A327259(2,12) = A327259(4,6) and 2, 4, 6 and 12 are in A327261.
72 is in the sequence because 72 = A327259(2,2,5) = A327259(6,6) and 2, 5 and 6 are in A327261. A327259(2,2,5) is well-defined because A327259(n,k) is associative.
221 is in the sequence because 221 = A327259(5,25) = A327259(11,11) and 5, 11 and 25 are in A327261.
462 is in the sequence because 462 = A327259(6,39) = A327259(11,22) = A327259(14,17) and 6, 11, 14, 17, 22 and 39 are in A327261.
The first six terms and their decompositions:
1 32 = A327259(2,2,2) = A327259(4,4)
2 48 = A327259(2,12) = A327259(4,6)
3 72 = A327259(2,2,5) = A327259(6,6)
4 96 = A327259(2,2,6) = A327259(4,12)
5 112 = A327259(2,28) = A327259(4,14)
6 126 = A327259(5,14) = A327259(6,11)
More in a-file.

Michael De Vlieger, Table of n, a(n) for n = 1..1256 (all terms m <= 10000)
David Lovler, Decomposition of a(n) into A327261(k)

Cf. A327259, A327261, A340747.

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]];F[d_]:=If[(q=Union[Sort/@(Position[Table[T[n,k],{n,2,Ceiling[d/3]},{k,2,Ceiling[d/3]}],d]+1)])=={},{{d}},q];FC[x_]:=FixedPoint[Union[Sort/@Flatten[Flatten/@Tuples[#]&/@((F/@#&/@#)&[#]),1]]&,F[x]];list={};Do[If[Length@FC@i>1,AppendTo[list,i]],{i,300}];list (* Giorgos Kalogeropoulos, Nov 05 2021 *)

Name amended by David Lovler, Jan 26 2022

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

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

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A340746 Numbers in array A322744 that do not have a unique decomposition into numbers of A307002 and are not equal to A322744(n,k), n > 1, k in the sequence.

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A348824 Numbers in array A327259 that do not have a unique decomposition into numbers of A327261.

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