A361748 -id:A361748 - cp's OEIS frontend

A361939 Inverse permutation to A361748.

1, 2, 4, 3, 7, 5, 11, 16, 6, 8, 22, 29, 37, 12, 9, 17, 46, 56, 67, 10, 13, 23, 79, 18, 92, 38, 106, 14, 121, 137, 154, 19, 24, 47, 15, 30, 172, 68, 39, 20, 191, 211, 232, 25, 254, 80, 277, 21, 301, 93, 48, 40, 326, 57, 26, 352, 69, 122, 379, 31, 407, 155, 436

Offset: 1

Rémy Sigrist, Mar 31 2023

nonn

			A361748(10) = 20, so a(20) = 10.

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A361748.

PARI
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See Links section.
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A364884 Triangle T(n, k), n > 0, k = 1..n, read by rows; T(1, 1) = 1, and for n > 1, k = 1..n, T(n, k) is the least unseen multiple of the k-th term of the flattened sequence .

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 7, 14, 16, 15, 18, 11, 20, 24, 21, 30, 32, 13, 22, 28, 27, 36, 40, 25, 17, 26, 44, 33, 42, 48, 35, 50, 19, 34, 52, 39, 54, 56, 45, 60, 72, 23, 38, 64, 51, 66, 80, 55, 70, 84, 63, 29, 46, 68, 57, 78, 88, 65, 90, 96, 81, 49

Offset: 1

Rémy Sigrist, Aug 12 2023

In other words, T(n, k) is a multiple of a(k).

As a flat sequence, this is a permutation of the positive integers (as each row starts with the least positive integer not yet in the sequence); see A364887 for the inverse permutation.

			Triangle begins:
          1,
          2,  4
          3,  6,  8
          5, 10, 12,  9
          7, 14, 16, 15,  18
         11, 20, 24, 21,  30,  32
         13, 22, 28, 27,  36,  40, 25
         17, 26, 44, 33,  42,  48, 35,  50
         19, 34, 52, 39,  54,  56, 45,  60,  72
         23, 38, 64, 51,  66,  80, 55,  70,  84, 63
         29, 46, 68, 57,  78,  88, 65,  90,  96, 81, 49
         31, 58, 76, 69, 102, 104, 75, 100, 108, 99, 77, 98
         ..................................................
    a(k)  1,  2,  4,  3,   6,   8,  5,  10,  12,  9,  7, 14

Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141 flattened)
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A361748 and A364885 for similar sequences.

Cf. A364887 (inverse).

PARI
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See Links section.
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A376905 Lexicographically earliest sequence of distinct positive integers with a(1) multiples of a number b(1) followed by a(2) multiples of a number b(2) etc.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 12, 5, 10, 15, 7, 14, 21, 28, 35, 42, 8, 16, 24, 32, 40, 48, 56, 64, 72, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 13, 26, 39, 52, 65, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 18, 36, 54, 90, 108, 126, 144, 162, 180, 198, 216

Offset: 1

Rémy Sigrist, Oct 08 2024

This sequence combines features of Golomb's sequence (A001462) and A361748.
This sequence is a permutation of the positive integers with inverse A376904.
This sequence can also be seen as an irregular table whose n-th row contains a(n) multiples of its leading term.

			The first terms/rows are:
  n  a(n)  b(n)  n-th row
  -  ----  ----  -------------------------------------------------
  1     1     1  1
  2     2     2  2, 4
  3     4     3  3, 6, 9, 12
  4     3     5  5, 10, 15
  5     6     7  7, 14, 21, 28, 35, 42
  6     9     8  8, 16, 24, 32, 40, 48, 56, 64, 72
  7    12    11  11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132
  8     5    13  13, 26, 39, 52, 65
  9    10    17  17, 34, 51, 68, 85, 102, 119, 136, 153, 170

Rémy Sigrist, Table of n, a(n) for n = 1..10063
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A001462, A361748, A376903, A376906 (inverse), A377093.

PARI
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\\ See Links section.
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from itertools import count, islice
def A376905gen(): # generator of terms
    aset, alst, m = {1, 2, 4}, [1, 2, 4], 3
    yield from [1, 2, 4]
    for n in count(3):
        nlst = []
        for k in count(m, m):
            if k not in aset:
                nlst.append(k)
                if len(nlst) == alst[n-1]:
                    break
        yield from nlst
        alst.extend(nlst)
        aset.update(nlst)
        while m in aset: m += 1
print(list(islice(A376905gen(), 70))) # Michael S. Branicky, Oct 16 2024

A377093 Irregular table read by rows; for any n > 0, let u(n) be the least positive integer not among the first n-1 rows; the n-th row lists the u(n) least multiples of u(n) not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 10, 15, 20, 25, 7, 14, 21, 28, 35, 42, 49, 8, 16, 24, 32, 40, 48, 56, 64, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 12, 36, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 169, 182

Offset: 1

Rémy Sigrist, Oct 16 2024

This sequence combines features of Golomb's sequence (A001462) and A361748.

This sequence is a permutation of the positive integers with inverse A377094.

			The first rows are:
    1;
    2, 4;
    3, 6, 9;
    5, 10, 15, 20, 25;
    7, 14, 21, 28, 35, 42, 49;
    8, 16, 24, 32, 40, 48, 56, 64;
    11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121;
    12, 36, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..10258
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A001462, A361748, A376903, A376905, A377094 (inverse).

PARI
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\\ See Links section.
```

from itertools import count, islice
def A377093gen(): # generator of terms
    aset, alst, m = {1, 2, 4}, [1, 2, 4], 3
    yield from [1, 2, 4]
    for n in count(3):
        nlst = []
        for k in count(m, m):
            if k not in aset:
                nlst.append(k)
                if len(nlst) == m:
                    break
        yield from nlst
        alst.extend(nlst)
        aset.update(nlst)
        while m in aset: m += 1
print(list(islice(A377093gen(), 70))) # Michael S. Branicky, Oct 16 2024

A361926 Square array A(n, k) of distinct positive integers, n, k > 0, read and filled by upwards antidiagonals in the greedy way such that A(n, k) is a multiple of A(n, 1).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 10, 9, 11, 14, 12, 16, 13, 15, 22, 21, 20, 18, 17, 19, 30, 33, 28, 24, 26, 23, 25, 38, 45, 44, 35, 32, 34, 27, 29, 50, 57, 60, 55, 42, 36, 40, 31, 37, 58, 75, 76, 90, 66, 49, 48, 46, 39, 41, 74, 87, 100, 95, 105, 77, 56, 52, 54, 43

Offset: 1

Rémy Sigrist, Mar 30 2023

This sequence is a variant of A361748.

As a flat sequence, this is a permutation of the positive integers with inverse A361940.

			Array A(n, k) begins:
  n\k |  1   2    3    4    5    6    7    8    9   10
  ----+-----------------------------------------------
    1 |  1   3    5    9   13   17   23   27   31   39
    2 |  2   6   10   16   18   26   34   40   46   54
    3 |  4   8   12   20   24   32   36   48   52   64
    4 |  7  14   21   28   35   42   49   56   63   70
    5 | 11  22   33   44   55   66   77   88   99  110
    6 | 15  30   45   60   90  105  120  135  165  180
    7 | 19  38   57   76   95  114  133  152  171  190
    8 | 25  50   75  100  125  150  175  200  225  250
    9 | 29  58   87  116  145  174  203  232  261  290
   10 | 37  74  111  148  185  222  259  296  333  370

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A361748, A361940 (inverse).

PARI
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See Links section.
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cp's OEIS Frontend

A361939 Inverse permutation to A361748.

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A364884 Triangle T(n, k), n > 0, k = 1..n, read by rows; T(1, 1) = 1, and for n > 1, k = 1..n, T(n, k) is the least unseen multiple of the k-th term of the flattened sequence .

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A376905 Lexicographically earliest sequence of distinct positive integers with a(1) multiples of a number b(1) followed by a(2) multiples of a number b(2) etc.

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A377093 Irregular table read by rows; for any n > 0, let u(n) be the least positive integer not among the first n-1 rows; the n-th row lists the u(n) least multiples of u(n) not yet in the sequence.

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PARI

Python

A361926 Square array A(n, k) of distinct positive integers, n, k > 0, read and filled by upwards antidiagonals in the greedy way such that A(n, k) is a multiple of A(n, 1).

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PARI