A337656 -id:A337656 - cp's OEIS frontend

A337657 Let M_k denote the addition table for the first k terms of A337656. 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.

5, 9, 11, 16, 17, 18, 22, 25, 26, 28, 29, 34, 35, 36, 38, 39, 41, 43, 46, 48, 49, 52, 53, 54, 55, 57, 58, 59, 61, 62, 63, 67, 69, 70, 71, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 90, 93, 96, 97, 99, 100, 101, 102, 104, 105, 106, 112, 113, 115, 116, 117, 118, 120, 124, 126, 128

Offset: 1

N. J. A. Sloane, Oct 01 2020

nonn

Note that if A337656(k+1) = t, all entries in M_{k+1} that are not entries in M_k are >= t.

			The addition table, M_9:
   + |  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

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A337657

Cf. A337655, A337656, A337658.

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.

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 *)

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).

Original entry on oeis.org

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 *)

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.

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.

A338012 a(1)=0; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n^2 different entries (the maximum possible).

Original entry on oeis.org

0, 3, 4, 10, 18, 23, 34, 55, 67, 93, 95, 120, 149, 166, 228, 271, 351, 398, 439, 505, 563, 611, 732, 771, 806, 924, 1052, 1121, 1278, 1412, 1586, 1654, 1875, 2012, 2245, 2341, 2445, 2616, 2819, 2920, 3034, 3322, 3518, 3754, 3918, 4016, 4311, 4649, 4848, 5321

Offset: 1

Peter Kagey, Oct 06 2020

nonn

			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..1000

Cf. A337655, A337656, A337946.

Cf. A338013, A338014, A338015, A338016, A338017.

cp's OEIS Frontend

A337657 Let M_k denote the addition table for the first k terms of A337656. 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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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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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

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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Mathematica

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

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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.

Views

Author

Keywords

Links

Crossrefs

A338012 a(1)=0; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n^2 different entries (the maximum possible).

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