A337660 -id:A337660 - 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.

A337661 Triangular array read by rows: T(n,k) = A337655(n) * A337655(k), for 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 03 2020

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

			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 rows, flattened)

Cf. A337655, A337659, A337660, A337662.

A338015 Addition table of A338012, read by antidiagonals.

Original entry on oeis.org

0, 3, 3, 4, 6, 4, 10, 7, 7, 10, 18, 13, 8, 13, 18, 23, 21, 14, 14, 21, 23, 34, 26, 22, 20, 22, 26, 34, 55, 37, 27, 28, 28, 27, 37, 55, 67, 58, 38, 33, 36, 33, 38, 58, 67, 93, 70, 59, 44, 41, 41, 44, 59, 70, 93, 95, 96, 71, 65, 52, 46, 52, 65, 71, 96, 95

Offset: 1

Peter Kagey, Oct 06 2020

			Addition table begins:
   + |  0  3  4 10 18 23  34  55  67
-----+-------------------------------
   0 |  0  3  4 10 18 23  34  55  67
   3 |  3  6  7 13 21 26  37  58  70
   4 |  4  7  8 14 22 27  38  59  71
  10 | 10 13 14 20 28 33  44  65  77
  18 | 18 21 22 28 36 41  52  73  85
  23 | 23 26 27 33 41 46  57  78  90
  34 | 34 37 38 44 52 57  68  89 101
  55 | 55 58 59 65 73 78  89 110 122
  67 | 67 70 71 77 85 90 101 122 134

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

Cf. A338012, A338014 (by rows), A338017 (multiplication table).

Cf. A337660, A337973.

A337973 Addition table from A337946, read by antidiagonals.

Original entry on oeis.org

2, 4, 4, 8, 6, 8, 13, 10, 10, 13, 23, 15, 14, 15, 23, 31, 25, 19, 19, 25, 31, 48, 33, 29, 24, 29, 33, 48, 62, 50, 37, 34, 34, 37, 50, 62, 86, 64, 54, 42, 44, 42, 54, 64, 86, 114, 88, 68, 59, 52, 52, 59, 68, 88, 114, 127, 116, 92, 73, 69, 60, 69, 73, 92, 116, 127

Offset: 1

Peter Kagey, Oct 05 2020

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

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

Cf. A337660, A337946, A338015, A338032.

Cf. A337972 (by rows), A337975 (multiplication).

A338032 Addition table of A337656, read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 7, 4, 4, 7, 12, 8, 6, 8, 12, 20, 13, 10, 10, 13, 20, 30, 21, 15, 14, 15, 21, 30, 44, 31, 23, 19, 19, 23, 31, 44, 65, 45, 33, 27, 24, 27, 33, 45, 65, 91, 66, 47, 37, 32, 32, 37, 47, 66, 91, 107, 92, 68, 51, 42, 40, 42, 51, 68, 92, 107

Offset: 1

Peter Kagey, Oct 07 2020

			Addition table begins:
   + |  0  1  3  7 12 20 30  44  65
-----+-----------------------------
   0 |  0  1  3  7 12 20 30  44  65
   1 |  1  2  4  8 13 21 31  45  66
   3 |  3  4  6 10 15 23 33  47  68
   7 |  7  8 10 14 19 27 37  51  72
  12 | 12 13 15 19 24 32 42  56  77
  20 | 20 21 23 27 32 40 50  64  85
  30 | 30 31 33 37 42 50 60  74  95
  44 | 44 45 47 51 56 64 74  88 109
  65 | 65 66 68 72 77 85 95 109 130

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

Cf. A337656, A337657, A338031 (by rows), A338034 (multiplication table).

Cf. A337660, A337973, A338015.

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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A337661 Triangular array read by rows: T(n,k) = A337655(n) * A337655(k), for 1 <= k <= n.

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A338015 Addition table of A338012, read by antidiagonals.

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A337973 Addition table from A337946, read by antidiagonals.

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A338032 Addition table of A337656, read by antidiagonals.

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