A088177 -id:A088177 - cp's OEIS frontend

A354759 The products of pairs of consecutive terms in A354749.

2, 6, 12, 20, 10, 14, 21, 15, 30, 42, 28, 36, 18, 22, 33, 24, 40, 35, 56, 72, 45, 55, 44, 52, 26, 34, 51, 39, 65, 60, 84, 63, 90, 70, 77, 66, 78, 91, 105, 120, 88, 99, 117, 104, 136, 68, 76, 38, 46, 69, 48, 80, 85, 102, 114, 57, 75, 50, 54, 108, 92, 115, 95, 133, 112, 144, 126, 154, 110, 130

Offset: 1

Scott R. Shannon, Jun 06 2022

nonn

See A354749 for further details.

			a(3) = 12 as A354749(3) * A354749(4) = 3 * 4 = 12.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A354749, A354803, A354804, A354753, A354754, A088177.

A354804 The products of consecutive terms in A354803.

Original entry on oeis.org

1, 2, 6, 3, 4, 12, 15, 5, 7, 14, 10, 20, 28, 21, 24, 8, 9, 18, 22, 11, 13, 26, 30, 60, 36, 45, 35, 42, 66, 33, 39, 52, 44, 55, 40, 56, 63, 72, 88, 77, 70, 90, 99, 110, 130, 65, 80, 16, 17, 34, 38, 19, 23, 46, 50, 25, 27, 54, 58, 29, 31, 62, 74, 37, 32, 96, 48, 112, 84, 132, 143, 78, 102

Offset: 1

Scott R. Shannon, Jun 07 2022

nonn

See A354803 for further details.

			a(6) = 12 as A354803(6) * A354803(7) = 4 * 3 = 12.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A354803, A354749, A354759, A354753, A354754, A088177.

A348439 Where prime(n) appears for the first time in A088178.

Original entry on oeis.org

2, 5, 6, 17, 18, 45, 46, 83, 84, 169, 170, 181, 182, 193, 194, 205, 206, 533, 534, 561, 562, 585, 586, 784, 785, 1040, 1041, 1068, 1069, 1096, 1097, 1126, 1127, 1150, 1151, 1932, 1933, 1986, 1987, 2022, 2023, 2062, 2063, 2090, 2091, 2118, 2119, 2146, 2147, 2178, 2179, 2206, 2207, 3929, 3930, 3965, 3966

Offset: 1

N. J. A. Sloane, Oct 19 2021

nonn

If n is odd, a(n) = A348438(n)-1; if n is even, a(n) = A348438(n).

N. J. A. Sloane, Table of n, a(n) for n = 1..91

Cf. A088177, A088178, A348438.

A317024 Lexicographically earliest sequence of positive terms such that for any distinct i and j, lcm(a(i), a(i+1)) and lcm(a(j), a(j+1)) are distinct.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 3, 5, 1, 7, 2, 5, 4, 7, 3, 8, 1, 9, 2, 11, 1, 13, 2, 15, 4, 9, 5, 7, 6, 11, 3, 13, 4, 11, 5, 8, 7, 9, 8, 11, 7, 10, 9, 11, 10, 13, 5, 16, 1, 17, 2, 19, 1, 23, 2, 25, 1, 27, 2, 29, 1, 31, 2, 32, 3, 16, 7, 12, 11, 13, 6, 17, 3, 19, 4, 17, 5, 19

Offset: 1

Rémy Sigrist, Jul 19 2018

nonn

See A317025 for the corresponding LCM.
This sequence has similarities with A088177.
For any n > 0, let g(n) = gcd(a(n), a(n+1)); between 1 and 800000, the function g takes only 5 times a value other than 1.
For any n > 0 and prime number p, if p divides a(n+1), then the p-adic valuation of a(n+1) is strictly greater than the p-adic valuation of a(n).
This sequence contains infinitely many distinct values.
The first occurrence of a prime number p, if not preceded by 1, is followed by 1.
The first occurrence of a prime power k, if not preceded by a divisor of k, is followed by 1.
If this sequence contains infinitely many 1's, then A317025 is a permutation of the natural numbers.

			The first terms, alongside lcm(a(n), a(n+1)), are:
  n  a(n)  lcm(a(n), a(n+1))
  -- ----  -----------------
   1    1    1
   2    1    2
   3    2    6
   4    3    3
   5    1    4
   6    4   12
   7    3   15
   8    5    5
   9    1    7
  10    7   14
  11    2   10
  12    5   20
  13    4   28
  14    7   21
  15    3   24
  16    8    8
  17    1    9
  18    9   18
  19    2   22
  20   11   11

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A317024
Rémy Sigrist, Scatterplot of the ordinal transform of the first 500000 terms

Cf. A088177, A317025.

PARI
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See Links section.
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A335943 Lexicographically earliest sequence of positive terms such that for any distinct m and n, the fractional parts of a(m)/a(m+1) and of a(n)/a(n+1) are distinct.

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 5, 4, 5, 6, 5, 7, 5, 8, 7, 6, 7, 8, 9, 7, 9, 8, 11, 7, 10, 7, 11, 8, 13, 9, 10, 9, 11, 9, 13, 10, 11, 10, 13, 11, 12, 11, 13, 12, 13, 14, 9, 14, 11, 14, 13, 15, 11, 15, 13, 16, 11, 16, 13, 17, 11, 17, 12, 17, 13, 18, 13, 19, 12, 19, 13, 20

Offset: 1

Rémy Sigrist, Jul 01 2020

nonn

For any k > 1, k appears up to A000010(k) times.

This sequence has similarities with A057979 and A088177, where we consider the ratio and the product of consecutive terms, respectively.

			The first terms, alongside the fractional part of a(n)/a(n+1), are:
  n   a(n)  frac(a(n)/a(n+1))
  --  ----  -----------------
   1     1          0
   2     1         1/2
   3     2         2/3
   4     3         3/4
   5     4         1/3
   6     3         3/5
   7     5         1/4
   8     4         4/5
   9     5         5/6
  10     6         1/5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of (n, frac(a(n)/a(n+1))) for n = 1..50000
Rémy Sigrist, Colored scatterplot of (numerator(frac(a(n)/a(n+1))), denominator(frac(a(n)/a(n+1)))) for n = 1..232289 (where the hue is function of n)
Rémy Sigrist, PARI program for A335943

See A335944 for a similar sequence.

Cf. A000010, A057979, A088177.

PARI
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See Links section.
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A348442 Records in A088178.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 15, 20, 24, 28, 32, 40, 42, 45, 48, 60, 70, 72, 78, 84, 98, 104, 108, 110, 120, 132, 143, 144, 147, 152, 160, 168, 189, 190, 198, 204, 209, 228, 234, 240, 256, 272, 280, 300, 304, 306, 323, 342, 360, 364, 378, 392, 420, 425, 450, 456, 460, 475, 500, 504, 506, 525, 550, 552, 575

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..3000

Cf. A088177, A088178, A348440-A348443.

from itertools import islice
def A348442(): # generator of terms
    yield 1
    c, p, a = 1, {1}, 1
    while True:
        n, na = 1, a
        while na in p:
            n += 1
            na += a
        p.add(na)
        a = n
        if c < na:
            c = na
            yield c
A348442_list = list(islice(A348442(),100)) # Chai Wah Wu, Oct 21 2021

A348443 Indices of records in A088178.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 11, 12, 14, 21, 22, 23, 27, 29, 33, 34, 36, 40, 53, 54, 55, 63, 65, 67, 69, 70, 72, 77, 93, 96, 99, 100, 101, 108, 111, 119, 121, 122, 124, 130, 131, 132, 140, 147, 149, 151, 153, 154, 156, 217, 237, 238, 239, 257, 258, 260, 263, 265, 266, 268, 272, 274, 275, 277, 279, 280

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..3000

Cf. A088177, A088178, A348440-A348442.

from itertools import islice
def A348443(): # generator of terms
    yield 1
    c, p, a, i = 1, {1}, 1, 1
    while True:
        n, na = 1, a
        while na in p:
            n += 1
            na += a
        p.add(na)
        a = n
        i += 1
        if c < na:
            c = na
            yield i
A348443_list = list(islice(A348443(),100)) # Chai Wah Wu, Oct 21 2021

A355072 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number whose sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=1..n-1, whose product a(n) * a(n-1) is distinct from all previous products, a(i) * a(i-1), i=1..n-1, and whose difference |a(n) - a(n-1)| is distinct from all previous differences, |a(i) - a(i-1)|, i=1..n-1.

Original entry on oeis.org

0, 1, 1, 3, 6, 1, 5, 11, 1, 9, 16, 1, 10, 21, 1, 13, 26, 1, 17, 3, 20, 1, 23, 5, 28, 1, 25, 46, 1, 29, 3, 32, 2, 34, 3, 37, 1, 40, 2, 42, 1, 44, 2, 46, 9, 42, 7, 53, 1, 49, 96, 2, 55, 4, 54, 103, 1, 61, 2, 59, 5, 60, 116, 1, 65, 2, 67, 1, 69, 7, 65, 126, 1, 72, 5, 74, 1, 73, 143, 1, 77, 3, 78, 155

Offset: 0

Scott R. Shannon, Jun 18 2022

nonn

For n up to ~35000 the vast majority of terms are concentrated along three lines, the lowest being near the x-axes; see the first linked image. In this same range there are many terms equal to 1; see A355135. Beyond this range the terms no longer fall along the upper-most line and the number of terms equal to 1 greatly diminishes. The reason for this change in behavior is unknown. The remaining upper-most line has a gradient close to 1 and contains multiple fixed points; see A355136 and the second linked image. The sequence it conjectured to contain all the positive integers.

			a(3) = 3 as a(2) = 1 and 3+1 = 4, 3*1 = 3, |3-1| = 2, and this product, sum, and difference has not occurred previously.
a(5) = 1 as a(4) = 6 and 1+6 = 7, 1*6 = 6, |1-6| = 5, and this product, sum, and difference has not occurred previously.

Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.
Scott R. Shannon, Image of the first 1000000 terms.

Cf. A355135, A355136, A088177, A008344, A110654.

A307838 Counterclockwise square spiral constructed by greedy algorithm such that the product of the values of any two vertically or horizontally adjacent cells is unique.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 2, 5, 7, 2, 11, 8, 3, 9, 5, 10, 2, 13, 7, 16, 3, 17, 5, 8, 18, 3, 19, 4, 13, 11, 5, 14, 7, 12, 8, 23, 1, 29, 5, 15, 8, 22, 5, 23, 4, 18, 17, 6, 26, 5, 27, 6, 25, 9, 11, 19, 1, 31, 9, 17, 11, 20, 7, 21, 8, 37, 3, 41, 11, 24, 9, 35, 6, 34, 10

Offset: 0

Rémy Sigrist, May 01 2019

nonn

This sequence is a two-dimensional variant of A088177.

Visually, we have a superposition of two images that we can separate by considering the parity of the sum of the x and y coordinates (see illustrations in Links section).

			The spiral begins:
    7---19---16---29---14---22---13---43----3---47----2
    |                                                 |
   31    8---21----7---20---11---17----9---31----1   43
    |    |                                       |    |
    2   37    1---23----8---12----7---14----5   19   14
    |    |    |                             |    |    |
   53    3   29    2---10----5----9----3   11   11   14
    |    |    |    |                   |    |    |    |
    4   41    5   13    3----3----2    8   13    9   21
    |    |    |    |    |         |    |    |    |    |
   37   11   15    7    4    1----1   11    4   25   11
    |    |    |    |    |              |    |    |    |
   10   24    8   16    2----5----7----2   19    6   31
    |    |    |    |                        |    |    |
   19    9   22    3---17----5----8---18----3   27   10
    |    |    |                                  |    |
   12   35    5---23----4---18---17----6---26----5   25
    |    |                                            |
   23    6---34---10---29---13----1---41----7---36----8
    |
    9---29----8---26---12---25---49----8---32---10---43

Rémy Sigrist, Table of n, a(n) for n = 0..10200 (-50 <= x <= 50 and -50 <= y <= 50)
Rémy Sigrist, Colored illustration of the sequence (with cells (x,y) such that -500 <= x <= 500 and -500 <= y <= 500)
Rémy Sigrist, Colored illustration of the sequence in function of the parities of x and y
Rémy Sigrist, PARI program for A307838

See A307834 for the additive variant.

Cf. A088177.

PARI
```
See Links section.
```

A330524 Lexicographically earliest sequence of positive terms such that for any distinct i and j, a(i) | a(j+1) <> a(j) | a(j+1) (where "|" corresponds to binary concatenation, A163621).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 3, 3, 4, 1, 4, 2, 4, 3, 5, 2, 5, 3, 6, 1, 8, 1, 9, 2, 8, 2, 9, 3, 7, 4, 4, 5, 4, 8, 3, 8, 4, 9, 4, 10, 2, 11, 2, 13, 1, 10, 4, 11, 3, 9, 5, 8, 5, 9, 6, 4, 15, 2, 16, 1, 16, 2, 17, 2, 18, 4, 16, 3, 10, 5, 10, 6, 5, 11, 4, 17, 3, 11, 5, 14

Offset: 1

Rémy Sigrist, Dec 17 2019

This sequence is a binary variant of A318225.
This sequence has similarities with A088177; here we combine successive terms by concatenation, there by multiplication.
This sequence is necessarily unbounded.
Also, the value 1 appears infinitely many times.

			The first terms, alongside their binary representation and that of the concatenation of two consecutive terms, are:
  n   a(n)  bin(a(n))  bin(a(n)|a(n+1))
  --  ----  ---------  ----------------
   1     1          1                11
   2     1          1               110
   3     2         10               101
   4     1          1               111
   5     3         11              1110
   6     2         10              1010
   7     2         10              1011
   8     3         11              1111
   9     3         11             11100
  10     4        100              1001
  11     1          1              1100
  12     4        100             10010

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A330525 for the concatenation of consecutive terms.

Cf. A088177, A163621, A318225.

s=0; v=1; for (n=1, 81, print1 (v", "); for (w=1, oo, if (!bittest(s, k=v*2^#binary(w)+w), s+=2^k; v=w; break)))

cp's OEIS Frontend

A354759 The products of pairs of consecutive terms in A354749.

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A354804 The products of consecutive terms in A354803.

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A348439 Where prime(n) appears for the first time in A088178.

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A317024 Lexicographically earliest sequence of positive terms such that for any distinct i and j, lcm(a(i), a(i+1)) and lcm(a(j), a(j+1)) are distinct.

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A335943 Lexicographically earliest sequence of positive terms such that for any distinct m and n, the fractional parts of a(m)/a(m+1) and of a(n)/a(n+1) are distinct.

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A348442 Records in A088178.

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A348443 Indices of records in A088178.

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A307838 Counterclockwise square spiral constructed by greedy algorithm such that the product of the values of any two vertically or horizontally adjacent cells is unique.

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A330524 Lexicographically earliest sequence of positive terms such that for any distinct i and j, a(i) | a(j+1) <> a(j) | a(j+1) (where "|" corresponds to binary concatenation, A163621).

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