A337659 -id:A337659 - 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 *)

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.

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.

A338014 Triangular array read by rows: T(n,k) = A338012(n) + A338012(k) for 1 <= k <= n.

Original entry on oeis.org

0, 3, 6, 4, 7, 8, 10, 13, 14, 20, 18, 21, 22, 28, 36, 23, 26, 27, 33, 41, 46, 34, 37, 38, 44, 52, 57, 68, 55, 58, 59, 65, 73, 78, 89, 110, 67, 70, 71, 77, 85, 90, 101, 122, 134, 93, 96, 97, 103, 111, 116, 127, 148, 160, 186, 95, 98, 99, 105, 113, 118, 129, 150, 162, 188, 190

Offset: 1

Peter Kagey, Oct 06 2020

			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

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

Cf. A338012, A338015 (by antidiagonals), A338016 (multiplication).

Cf. A337659, A337972.

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).

A338031 Triangular array read by rows: T(n,k) = A337656(n) + A337656(k) for 1 <= k <= n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 14, 12, 13, 15, 19, 24, 20, 21, 23, 27, 32, 40, 30, 31, 33, 37, 42, 50, 60, 44, 45, 47, 51, 56, 64, 74, 88, 65, 66, 68, 72, 77, 85, 95, 109, 130, 91, 92, 94, 98, 103, 111, 121, 135, 156, 182, 107, 108, 110, 114, 119, 127, 137, 151, 172, 198, 214

Offset: 1

Peter Kagey, Oct 07 2020

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

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

Cf. A337656, A337657, A338032 (by antidiagonals), A338033 (multiplication).

Cf. A337659, A337972, A338014.

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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A337661 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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A338014 Triangular array read by rows: T(n,k) = A338012(n) + A338012(k) for 1 <= k <= n.

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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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A338031 Triangular array read by rows: T(n,k) = A337656(n) + A337656(k) for 1 <= k <= n.

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