A337946 -id:A337946 - cp's OEIS frontend

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

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.

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).

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).

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).

A337975 Multiplication table from A337946, read by antidiagonals.

Original entry on oeis.org

1, 3, 3, 7, 9, 7, 12, 21, 21, 12, 22, 36, 49, 36, 22, 30, 66, 84, 84, 66, 30, 47, 90, 154, 144, 154, 90, 47, 61, 141, 210, 264, 264, 210, 141, 61, 85, 183, 329, 360, 484, 360, 329, 183, 85, 113, 255, 427, 564, 660, 660, 564, 427, 255, 113, 126, 339, 595, 732, 1034, 900, 1034, 732, 595, 339, 126

Offset: 1

Peter Kagey, Oct 05 2020

			Multiplication table for A337946 begins:
  *  |  1   3   7   12   22   30   47   61   85
-----+-----------------------------------------
   1 |  1   3   7   12   22   30   47   61   85
   3 |  3   9  21   36   66   90  141  183  255
   7 |  7  21  49   84  154  210  329  427  595
  12 | 12  36  84  144  264  360  564  732 1020
  22 | 22  66 154  264  484  660 1034 1342 1870
  30 | 30  90 210  360  660  900 1410 1830 2550
  47 | 47 141 329  564 1034 1410 2209 2867 3995
  61 | 61 183 427  732 1342 1830 2867 3721 5185
  85 | 85 255 595 1020 1870 2550 3995 5185 7225

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

Cf. A337662, A337946, A338017, A338034.

Cf. A337973 (addition), A337974 (read by rows).

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 *)

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.

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

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

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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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A337973 Addition table from A337946, read by antidiagonals.

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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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A337975 Multiplication table from A337946, 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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A337659 Triangular array read by rows: T(n,k) = A337655(n) + A337655(k), for 1 <= k <= n.

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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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