A307002 -id:A307002 - cp's OEIS frontend

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

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

David Lovler, Jan 20 2021

nonn

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.

			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.

David Lovler, The first 49 terms and their decompositions in A322744(n,k).

Cf. A307002, A322744, A340746.

A340748 Numbers m > 3 such that m-1, m, m+1 belong to A307002.

Original entry on oeis.org

4, 22, 94, 142, 262, 334, 358, 694, 862, 934, 1174, 1678, 1822, 2182, 2854, 3022, 3862, 3958, 4054, 4702, 4894, 5062, 5398, 5854, 6022, 6238, 6382, 6694, 7534, 7558, 7822, 8038, 8422, 9502, 9934, 10078, 10342, 10558, 11062, 11758, 12574, 12622, 13942, 14038, 14254, 14374, 15094, 16438, 16462

Offset: 1

David Lovler, Jan 30 2021

nonn

All terms m == 1 (mod 3). To prove this, look at gaps of 3 in row 2 of array A322744(n,k). The longest strings of consecutive numbers not in A322744(n,k) can occur only for these 3 numbers. The number following such a gap is A322744(2,e) = (3*2*e)/2 = 3e for some even e. The middle of the string, m = A322744(2,e) - 2 = 3e - 2. Thus m == 1 (mod 3). After the first term, all terms m == 2 (mod 4).- David Lovler, Nov 29 2021

David Lovler, Table of n, a(n) for n = 1..852

Cf. A307002, A322744, A345357, A345473, A345474.

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
list(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, T(n, k)); ); ); setminus([1..nn], Set(list)); } \\ A307002
lista(nn) = {my(v=Vec(list(nn))); for (m=4, #v-1, my(x=v[m]); if (vecsearch(v, x-1) && vecsearch(v, x+1), print1(x, ", ")););} \\ Michel Marcus, Apr 02 2021

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 *)

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

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

Original entry on oeis.org

3, 5, 2, 9, 3, 2, 13, 5, 3, 2, 21, 7, 4, 3, 2, 25, 11, 5, 4, 3, 2, 33, 13, 7, 5, 4, 3, 2, 37, 17, 9, 6, 5, 4, 3, 2, 45, 19, 10, 7, 6, 5, 4, 3, 2, 57, 23, 13, 9, 7, 6, 5, 4, 3, 2, 61, 29, 15, 11, 8, 7, 6, 5, 4, 3, 2, 73, 31, 17, 12, 9, 8, 7, 6, 5, 4, 3, 2

Offset: 1

David Lovler, Oct 15 2019

All the U(i;n,k) mimic the ordinary multiplication table in that they are commutative, associative, have identity element 1 and have 0. However (except when i=2) they are partially distributive, meaning that distributivity works except if an even number is partitioned into a sum of two odd numbers. Only when i=2, the odd-even-dependent A319929 term disappears and normal distributivity holds.
U(0;n,k) = A319929(n,k);
U(1;n,k) = A322630(n,k);
U(2;n,k) = n*k;
U(3;n,k) = A322744(n,k);
U(4;n,k) = A327259(n,k);
U(i;n,k) = 2i*floor(n/2)*floor(k/2) + A319929(n,k).
Row 1 is 2p-1 where p is a prime number (A076274 without 1).
Row 2 is the prime numbers.
Row 3 is A307002.
Row 4 is A327261.
The i-th row of T(n,k) consists of numbers that sieve out of the array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2, in numerical order.
From David Lovler, Sep 02 2020: (Start)
Row 1 has no even numbers. Row 2 has one even number. Generally, the even numbers of the i-th row start with i-1 consecutive even numbers (from 2). This is because U(i;2,2) = 2*i gives the first even number not in row i.
Row 3 seems to have even numbers that, after 2, coincide with A112774 which has an infinite number of terms. For i > 3, as i increases, row i has a denser presence of even numbers, thus each row has an infinite number of even terms.
Generalization of the twin prime conjecture: Since row 2 is the prime numbers, we can observe the twin prime conjecture that after the first three odd primes, the sprinkling of pairs of consecutive prime numbers never ends. Concerning just odd terms, a similar conjecture can be stated for rows i >= 3. Row 3 starts with four odd numbers then the sprinkling of three consecutive odd number never ends. Row 4 starts with five odd numbers then the sprinkling of four consecutive odd numbers never ends. The pattern continues as row i starts with i+1 odd numbers then the sprinkling of i consecutive odd numbers never ends. We can take this back to row 1 which starts with two odd numbers then continues with isolated odd numbers.
Studying the even terms, there is an analog to the above generalization of the twin prime conjecture. Row 3 starts with two even numbers then continues with isolated even numbers. Row 4 starts with three even numbers then the sprinkling of pairs of consecutive even numbers never ends. Row 5 starts with four even numbers then the sprinkling of three consecutive even numbers never ends. The pattern continues as row i starts with i-1 even numbers then the sprinkling of i-2 consecutive even numbers never ends.
(End)

			3  5  9  13  21  25  33  37  45  57  61  73  81  85  93 105 117 121 133 141 145 ...
2  3  5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73 ...
2  3  4   5   7   9  10  13  15  17  21  22  23  25  29  31  34  37  39  41  45 ...
2  3  4   5   6   7   9  11  12  14  15  17  19  21  22  25  27  28  29  31  35 ...
2  3  4   5   6   7   8   9  11  13  14  16  17  18  19  21  23  25  26  28  29 ...
2  3  4   5   6   7   8   9  10  11  13  15  16  18  19  20  21  22  23  25  27 ...
2  3  4   5   6   7   8   9  10  11  12  13  15  17  18  20  21  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  17  19  20  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  19  21  22  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  21  23  24 ...
...

Cf. A000040, A003991, A076274, A112774.
Cf. A319929, A322630, A322744, A327259, A345474.
Cf. A354596.

row=12;max=200;U[i_,n_,k_]:=(i*n*k-(i-2)If[OddQ@n,If[OddQ@k,n+k-1,k],If[OddQ@k,n,0]])/2;t=Table[c=Union@Flatten@Table[U[i,n,k],{n,2,max},{k,2,max}];Complement[Range[2,max],c][[;;row]],{i,row}];Flatten@Table[t[[m,k-m+1]],{k,row},{m,k}] (* Giorgos Kalogeropoulos, Jun 08 2021 *)

With one exception there are likely no formulas for the rows of T(n,k) since their creation is based on a sieving process like the familiar prime number sieve. The exception is T(1,k) = 2*T(2,k)-1.

A307001 Odd numbers > 1 not of the form (3n*k - n - k + 1)/2 where n and k are odd numbers > 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 15, 17, 21, 23, 25, 29, 31, 37, 39, 41, 45, 49, 53, 55, 57, 63, 65, 69, 71, 73, 77, 79, 81, 85, 93, 95, 97, 101, 105, 109, 111, 119, 121, 125, 129, 133, 135, 137, 141, 143, 149, 151, 153, 157, 161, 169, 175, 177, 181, 183, 185, 189, 193, 197

Offset: 1

David Lovler, Mar 19 2019

Terms are the odd numbers > 1 not appearing in array A322744 with its first row and column omitted.

They are the odd numbers in A307002. - David Lovler, Jan 17 2022

Cf. A322744, A307002.

Select[2 Range[100]-1, FindInstance[# == 1 + 2*n + k (2 + 6 n) && n>0 && k>0, {n,k}, Integers] === {} &] (* Giovanni Resta, May 06 2019 *)

isok(n) = {my(kj, tij); if (n % 2, forstep (i=3, oo, 2, kj = 0; forstep (j=3, i, 2, tij = (3*i*j - i - j +1)/2; if (tij == n, return (0)); if (tij > n, kj = j; break);); if ((kj == 3) && (tij > n), break);); return (n>1));} \\ Michel Marcus, Apr 24 2019 and Jan 25 2022

Definition amended by David Lovler, Jan 25 2022

cp's OEIS Frontend

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

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A340748 Numbers m > 3 such that m-1, m, m+1 belong to A307002.

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

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A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

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A307001 Odd numbers > 1 not of the form (3n*k - n - k + 1)/2 where n and k are odd numbers > 1.

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

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.