A337661 -id:A337661 - cp's OEIS frontend

A337655 a(1)=1; thereafter, a(n) is the smallest number such that both the addition and multiplication tables for (a(1),...,a(n)) contain n*(n+1)/2 different entries (the maximum possible).

1, 2, 5, 7, 15, 22, 31, 50, 68, 90, 101, 124, 163, 188, 215, 253, 322, 358, 455, 486, 527, 631, 702, 780, 838, 920, 1030, 1062, 1197, 1289, 1420, 1500, 1689, 1765, 1886, 2114, 2353, 2410, 2570, 2686, 2857, 3063, 3207, 3477, 3616, 3845, 3951, 4150, 4480, 4595, 4746, 5030, 5286, 5698, 5999, 6497, 6624, 6938, 7219, 7661, 7838, 8469, 8665, 9198, 9351, 9667, 9966

Offset: 1

Jean-Paul Delahaye, Sep 30 2020

nonn

If one specifies that not only are there n(n+1)/2 distinct numbers in the addition and multiplication tables, but that all n(n+1) numbers are distinct, then the sequence is A337946 - David A. Corneth, Oct 02 2020

Rémy Sigrist, Table of n, a(n) for n = 1..3000 (a(1)-a(101) from Jean-Paul Delahaye, a(102)-a(1000) from Peter Kagey)
Rémy Sigrist, C++ program for A337655

See A337659 and A337660 (for the addition table), and A337661 and A337662 (for the multiplication table).
Cf. A337656, A337657, A337658.
For similar sequences that focus just on the addition or multiplication tables, see A005282 and A066720.
Cf. also A337946.

terms=67;a[1]=b[1]=1;a1=b1={1};Do[k=a[n-1]+1;While[a2=Union@Join[{2k},Array[a@#+k&,n-1]];b2=Union@Join[{k^2},Array[b@#*k&,n-1]];Intersection[a2,a1]!={}||Intersection[b2,b1]!={},k++];a[n]=b[n]=k;a1=Union[a1,a2];b1=Union[b1,b2],{n,2,terms}];Array[a,terms] (* Giorgos Kalogeropoulos, Nov 15 2021 *)

A337659 Triangular array read by rows: T(n,k) = A337655(n) + A337655(k), for 1 <= k <= n.

Original entry on oeis.org

2, 3, 4, 6, 7, 10, 8, 9, 12, 14, 16, 17, 20, 22, 30, 23, 24, 27, 29, 37, 44, 32, 33, 36, 38, 46, 53, 62, 51, 52, 55, 57, 65, 72, 81, 100, 69, 70, 73, 75, 83, 90, 99, 118, 136, 91, 92, 95, 97, 105, 112, 121, 140, 158, 180, 102, 103, 106, 108, 116, 123, 132, 151, 169, 191, 202

Offset: 1

N. J. A. Sloane, Oct 03 2020

This is the lower triangular part of the addition table from A337655, read by rows.

Sequences A337659, A337660, A337661, A337662 arise from the addition and multiplication tables in A337655, each one described in two ways. Perhaps someone could help by creating the analogous sets of four sequences for the addition and multiplication tables in the closely related sequences A337656 and A337946.

			The addition table from A337655 begins:
   2,  3,  6,  8,  16,  23,  32,  51,  69,  91, ...
   3,  4,  7,  9,  17,  24,  33,  52,  70,  92, ...
   6,  7, 10, 12,  20,  27,  36,  55,  73,  95, ...
   8,  9, 12, 14,  22,  29,  38,  57,  75,  97, ...
  16, 17, 20, 22,  30,  37,  46,  65,  83, 105, ...
  23, 24, 27, 29,  37,  44,  53,  72,  90, 112, ...
  32, 33, 36, 38,  46,  53,  62,  81,  99, 121, ...
  51, 52, 55, 57,  65,  72,  81, 100, 118, 140, ...
  69, 70, 73, 75,  83,  90,  99, 118, 136, 158, ...
  91, 92, 95, 97, 105, 112, 121, 140, 158, 180, ...
  ...

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

Cf. A337655, A337660, A337661, A337662.

A337662 Multiplication table from A337655, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 5, 4, 5, 7, 10, 10, 7, 15, 14, 25, 14, 15, 22, 30, 35, 35, 30, 22, 31, 44, 75, 49, 75, 44, 31, 50, 62, 110, 105, 105, 110, 62, 50, 68, 100, 155, 154, 225, 154, 155, 100, 68, 90, 136, 250, 217, 330, 330, 217, 250, 136, 90, 101, 180, 340, 350, 465, 484, 465, 350, 340, 180

Offset: 1

N. J. A. Sloane, Oct 04 2020

			The multiplication table from A337655 begins:
   1,   2,   5,   7,   15,   22,   31,   50,   68,   90, ...
   2,   4,  10,  14,   30,   44,   62,  100,  136,  180, ...
   5,  10,  25,  35,   75,  110,  155,  250,  340,  450, ...
   7,  14,  35,  49,  105,  154,  217,  350,  476,  630, ...
  15,  30,  75, 105,  225,  330,  465,  750, 1020, 1350, ...
  22,  44, 110, 154,  330,  484,  682, 1100, 1496, 1980, ...
  31,  62, 155, 217,  465,  682,  961, 1550, 2108, 2790, ...
  50, 100, 250, 350,  750, 1100, 1550, 2500, 3400, 4500, ...
  68, 136, 340, 476, 1020, 1496, 2108, 3400, 4624, 6120, ...
  90, 180, 450, 630, 1350, 1980, 2790, 4500, 6120, 8100, ...
  ...

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

Cf. A337655, A337659, A337661, A337662.

A337660 Addition table from A337655, read by antidiagonals.

Original entry on oeis.org

2, 3, 3, 6, 4, 6, 8, 7, 7, 8, 16, 9, 10, 9, 16, 23, 17, 12, 12, 17, 23, 32, 24, 20, 14, 20, 24, 32, 51, 33, 27, 22, 22, 27, 33, 51, 69, 52, 36, 29, 30, 29, 36, 52, 69, 91, 70, 55, 38, 37, 37, 38, 55, 70, 91, 102, 92, 73, 57, 46, 44, 46, 57, 73, 92, 102, 125, 103, 95, 75, 65, 53, 53, 65, 75, 95, 103, 125

Offset: 1

N. J. A. Sloane, Oct 04 2020

			The addition table from A337655 begins:
   2,  3,  6,  8,  16,  23,  32,  51,  69,  91, ...
   3,  4,  7,  9,  17,  24,  33,  52,  70,  92, ...
   6,  7, 10, 12,  20,  27,  36,  55,  73,  95, ...
   8,  9, 12, 14,  22,  29,  38,  57,  75,  97, ...
  16, 17, 20, 22,  30,  37,  46,  65,  83, 105, ...
  23, 24, 27, 29,  37,  44,  53,  72,  90, 112, ...
  32, 33, 36, 38,  46,  53,  62,  81,  99, 121, ...
  51, 52, 55, 57,  65,  72,  81, 100, 118, 140, ...
  69, 70, 73, 75,  83,  90,  99, 118, 136, 158, ...
  91, 92, 95, 97, 105, 112, 121, 140, 158, 180, ...
  ...

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

Cf. A337655, A337659, A337661, A337662.

A338016 Triangular array read by rows: T(n,k) = A338012(n) * A338012(k) for 1 <= k <= n.

Original entry on oeis.org

0, 0, 9, 0, 12, 16, 0, 30, 40, 100, 0, 54, 72, 180, 324, 0, 69, 92, 230, 414, 529, 0, 102, 136, 340, 612, 782, 1156, 0, 165, 220, 550, 990, 1265, 1870, 3025, 0, 201, 268, 670, 1206, 1541, 2278, 3685, 4489, 0, 279, 372, 930, 1674, 2139, 3162, 5115, 6231, 8649

Offset: 1

Peter Kagey, Oct 06 2020

			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..10011 (first 141 rows, flattened)

Cf. A338012, A338014 (addition), A338017 (by antidiagonals).

Cf. A337661, A337974.

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).

A338033 Triangular array read by rows: T(n,k) = A337656(n) * A337656(k) for 1 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 3, 9, 0, 7, 21, 49, 0, 12, 36, 84, 144, 0, 20, 60, 140, 240, 400, 0, 30, 90, 210, 360, 600, 900, 0, 44, 132, 308, 528, 880, 1320, 1936, 0, 65, 195, 455, 780, 1300, 1950, 2860, 4225, 0, 91, 273, 637, 1092, 1820, 2730, 4004, 5915, 8281

Offset: 1

Peter Kagey, Oct 07 2020

			Multiplication table begins:
   * | 0  1   3   7  12   20   30   44   65
-----+-------------------------------------
   0 | 0
   1 | 0  1
   3 | 0  3   9
   7 | 0  7  21  49
  12 | 0 12  36  84 144
  20 | 0 20  60 140 240  400
  30 | 0 30  90 210 360  600  900
  44 | 0 44 132 308 528  880 1320 1936
  65 | 0 65 195 455 780 1300 1950 2860 4225

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

Cf. A337656, A337657, A338031 (addition), A338034 (by antidiagonals).

Cf. A337661, A337974, A338016.

cp's OEIS Frontend

A337655 a(1)=1; thereafter, a(n) is the smallest number such that both the addition and multiplication tables for (a(1),...,a(n)) contain n*(n+1)/2 different entries (the maximum possible).

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A337659 Triangular array read by rows: T(n,k) = A337655(n) + A337655(k), for 1 <= k <= n.

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A337662 Multiplication table from A337655, read by antidiagonals.

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A337660 Addition table from A337655, read by antidiagonals.

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A338016 Triangular array read by rows: T(n,k) = A338012(n) * A338012(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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A338033 Triangular array read by rows: T(n,k) = A337656(n) * A337656(k) for 1 <= k <= n.

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