A337657 -id:A337657 - cp's OEIS frontend

A337656 a(1)=0; thereafter, a(n) is the smallest number such that the addition table for (a(1),...,a(n)) contains n(n+1)/2 different entries, and the multiplication table for (a(2),...,a(n)) contains n(n-1)/2 different nonzero entries (the maximum possible in both cases).

0, 1, 3, 7, 12, 20, 30, 44, 65, 91, 107, 122, 147, 196, 230, 263, 329, 375, 397, 472, 545, 596, 677, 716, 770, 931, 1007, 1059, 1201, 1330, 1415, 1484, 1570, 1688, 1934, 1982, 2229, 2391, 2550, 2646, 2837, 3078, 3138, 3208, 3383, 3560, 3596, 3930, 4257, 4421, 4673

Offset: 1

N. J. A. Sloane, Oct 01 2020

nonn

Inspired by A337655.

The distinct entries in the addition table for a(1),...,a(n), when sorted into increasing order, converge to a sequence which consists of all positive numbers except 5, 9, 11, 16, 17, 18, 22, 25, 26, 28, ... (see A337657).

Rémy Sigrist, Table of n, a(n) for n = 1..3000 (first 87 terms from N. J. A. Sloane)
Rémy Sigrist, C++ program for A337656

Cf. A337655, A337657, A337658.

Corrected and extended by Jean-Paul Delahaye, Oct 01 2020

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).

Original entry on oeis.org

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 *)

A337658 Let M_k denote the addition table for the first k terms of A337655. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

Original entry on oeis.org

1, 5, 11, 13, 15, 18, 19, 21, 25, 26, 28, 31, 34, 35, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 54, 56, 58, 59, 60, 61, 63, 64, 66, 67, 68, 71, 74, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 93, 94, 96, 98, 101, 104, 107, 109, 110, 111, 113, 114, 115, 117, 119, 120, 122, 124, 127

Offset: 1

Jean-Paul Delahaye, Oct 01 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A337658

Cf. A337655, A337656, A337657.

A337947 Numbers that do not appear in the addition or multiplication table of A337946.

Original entry on oeis.org

5, 11, 16, 17, 18, 20, 26, 27, 28, 32, 35, 38, 39, 40, 41, 43, 45, 46, 51, 53, 55, 56, 57, 58, 63, 65, 67, 70, 71, 72, 74, 75, 76, 78, 79, 80, 81, 82, 87, 89, 93, 95, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 109, 110, 111, 112, 117, 118, 119, 121, 123

Offset: 1

Peter Kagey, Oct 02 2020

nonn

			The addition table of A337946(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 A337946(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
Neither of these two tables contain 5, 11, 16, 17, 18, or 20. Since each row is strictly increasing, the rest of the two (infinite) tables do not contain these values either.

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

Cf. A263995, A337657, A337658, A337946.

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.

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.

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.

A338034 Multiplication table of A337656, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 7, 9, 7, 0, 0, 12, 21, 21, 12, 0, 0, 20, 36, 49, 36, 20, 0, 0, 30, 60, 84, 84, 60, 30, 0, 0, 44, 90, 140, 144, 140, 90, 44, 0, 0, 65, 132, 210, 240, 240, 210, 132, 65, 0, 0, 91, 195, 308, 360, 400, 360, 308, 195, 91, 0

Offset: 1

Peter Kagey, Oct 07 2020

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

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

Cf. A337656, A337657, A338032 (addition table), A338033 (by rows).

Cf. A337662, A337975, A338017.

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

A337656 a(1)=0; thereafter, a(n) is the smallest number such that the addition table for (a(1),...,a(n)) contains n*(n+1)/2 different entries, and the multiplication table for (a(2),...,a(n)) contains n*(n-1)/2 different nonzero entries (the maximum possible in both cases).

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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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Mathematica

A337658 Let M_k denote the addition table for the first k terms of A337655. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

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Crossrefs

A337947 Numbers that do not appear in the addition or multiplication table of A337946.

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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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A338032 Addition table of A337656, read by antidiagonals.

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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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A338034 Multiplication table of A337656, read by antidiagonals.

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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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A337656 a(1)=0; thereafter, a(n) is the smallest number such that the addition table for (a(1),...,a(n)) contains n(n+1)/2 different entries, and the multiplication table for (a(2),...,a(n)) contains n(n-1)/2 different nonzero entries (the maximum possible in both cases).