A337947 -id:A337947 - cp's OEIS frontend

A337946 a(1)=1; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n*(n+1) different entries (the maximum possible).

1, 3, 7, 12, 22, 30, 47, 61, 85, 113, 126, 177, 193, 246, 279, 321, 341, 428, 499, 571, 616, 686, 754, 854, 975, 1052, 1150, 1317, 1376, 1457, 1513, 1664, 1761, 1961, 2307, 2434, 2591, 2795, 2843, 3057, 3226, 3405, 3508, 3776, 3930, 4023, 4196, 4575, 4731

Offset: 1

Peter Kagey, Oct 02 2020

nonn

			The addition table of a(k) for k=1..5:
   + | 1 3  7 12 22
  ---+-------------
   1 | 2 4  8 13 23
   3 |   6 10 15 25
   7 |     14 19 29
  12 |        24 34
  22 |           44
The multiplication table of a(k) for k=1..5:
   * | 1 3  7  12  22
  ---+---------------
   1 | 1 3  7  12  22
   3 |   9 21  36  66
   7 |     49  84 154
  12 |        144 264
  22 |            484
These two tables contain the 5*(5+1) = 30 values {1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 19, 21, 22, 23, 24, 25, 29, 34, 36, 44, 49, 66, 84, 144, 154, 264, 484}.

Peter Kagey, Table of n, a(n) for n = 1..1000
Peter Kagey, Haskell program

Cf. A005282 (addition table), A066720 (multiplication table), A337655, A337656, A337947.

j={k=1};Do[While[l=Join[j,{++k}];g=Union[Sort/@Tuples[l,{2}]];p=Times@@#&/@g;s=Total/@g;!SameQ@@Flatten[{Length@Union@Flatten@{p,s},Length@l(Length@l+1)}]];j=Join[j,{k}];k=Last@j,48];j (* Giorgos Kalogeropoulos, Nov 16 2021 *)

A337972 Triangular array read by rows: T(n,k) = A337946(n) + A337946(k), for 1 <= k <= n.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 13, 15, 19, 24, 23, 25, 29, 34, 44, 31, 33, 37, 42, 52, 60, 48, 50, 54, 59, 69, 77, 94, 62, 64, 68, 73, 83, 91, 108, 122, 86, 88, 92, 97, 107, 115, 132, 146, 170, 114, 116, 120, 125, 135, 143, 160, 174, 198, 226, 127, 129, 133, 138, 148, 156, 173, 187, 211, 239, 252

Offset: 1

Peter Kagey, Oct 05 2020

This sequence, A337947, and A337974 partition the positive integers.

			Addition table for A337946 begins:
  +  |  1  3  7 12  22  30  47  61  85
-----+--------------------------------
   1 |  2
   3 |  4  6
   7 |  8 10 14
  12 | 13 15 19 24
  22 | 23 25 29 34  44
  30 | 31 33 37 42  52  60
  47 | 48 50 54 59  69  77  94
  61 | 62 64 68 73  83  91 108 122
  85 | 86 88 92 97 107 115 132 146 170

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A337659, A337946, A338014, A338031.

Cf. A337973 (read by antidiagonals), A337974 (multiplication table).

A337974 Triangular array read by rows: T(n,k) = A337946(n) * A337946(k), for 1 <= k <= n.

Original entry on oeis.org

1, 3, 9, 7, 21, 49, 12, 36, 84, 144, 22, 66, 154, 264, 484, 30, 90, 210, 360, 660, 900, 47, 141, 329, 564, 1034, 1410, 2209, 61, 183, 427, 732, 1342, 1830, 2867, 3721, 85, 255, 595, 1020, 1870, 2550, 3995, 5185, 7225, 113, 339, 791, 1356, 2486, 3390, 5311, 6893, 9605, 12769

Offset: 1

Peter Kagey, Oct 05 2020

This sequence, A337947, and A337972 partition the positive integers.

			Multiplication table for A337946 begins:
  *  |  1   3   7   12   22   30   47   61   85
-----+-----------------------------------------
   1 |  1
   3 |  3   9
   7 |  7  21  49
  12 | 12  36  84  144
  22 | 22  66 154  264  484
  30 | 30  90 210  360  660  900
  47 | 47 141 329  564 1034 1410 2209
  61 | 61 183 427  732 1342 1830 2867 3721
  85 | 85 255 595 1020 1870 2550 3995 5185 7225

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A337661, A337946, A338016, A338033.

Cf. A337972 (addition table), A337975 (read by antidiagonals).

A338013 The list of positive integers that do not appear in the addition or multiplication tables of A338012.

Original entry on oeis.org

1, 2, 5, 11, 15, 17, 19, 24, 25, 29, 31, 32, 35, 39, 42, 43, 45, 47, 48, 49, 50, 51, 53, 56, 60, 61, 62, 63, 64, 66, 74, 75, 76, 79, 80, 81, 82, 83, 84, 86, 87, 88, 91, 94, 104, 106, 107, 108, 109, 112, 114, 115, 117, 119, 121, 125, 126, 128, 131, 132, 133

Offset: 1

Peter Kagey, Oct 06 2020

nonn

			The first four terms of this sequence are 1, 2, 5, and 11 because these are the four smallest positive integers that do not appear in the addition and multiplication tables of A338012.
Addition table begins:
   + |  0  3  4 10 18 23  34  55  67
-----+-------------------------------
   0 |  0
   3 |  3  6
   4 |  4  7  8
  10 | 10 13 14 20
  18 | 18 21 22 28 36
  23 | 23 26 27 33 41 46
  34 | 34 37 38 44 52 57  68
  55 | 55 58 59 65 73 78  89 110
  67 | 67 70 71 77 85 90 101 122 134
Multiplication table begins:
   * | 0   3   4  10   18   23   34   55   67
-----+---------------------------------------
   0 | 0
   3 | 0   9
   4 | 0  12  16
  10 | 0  30  40 100
  18 | 0  54  72 180  324
  23 | 0  69  92 230  414  529
  34 | 0 102 136 340  612  782 1156
  55 | 0 165 220 550  990 1265 1870 3025
  67 | 0 201 268 670 1206 1541 2278 3685 4489

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Haskell Program.

Cf. A338012.

Cf. A337657, A337658, A337947.

cp's OEIS Frontend

A337946 a(1)=1; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n*(n+1) different entries (the maximum possible).

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A337972 Triangular array read by rows: T(n,k) = A337946(n) + A337946(k), for 1 <= k <= n.

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A337974 Triangular array read by rows: T(n,k) = A337946(n) * A337946(k), for 1 <= k <= n.

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A338013 The list of positive integers that do not appear in the addition or multiplication tables of A338012.

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