A088177 -id:A088177 - cp's OEIS frontend

A334857 Lexicographically earliest sequence of integers > 1 such that for any distinct m and n, a(m)^a(m+1) <> a(n)^a(n+1).

2, 2, 3, 2, 4, 3, 3, 4, 4, 5, 2, 5, 3, 5, 4, 6, 2, 7, 2, 9, 3, 7, 3, 8, 5, 5, 6, 3, 9, 5, 7, 4, 7, 5, 8, 6, 4, 8, 7, 6, 5, 9, 6, 6, 7, 7, 8, 8, 9, 7, 9, 8, 10, 2, 11, 2, 13, 2, 17, 2, 19, 2, 20, 2, 22, 2, 23, 2, 25, 5, 11, 3, 11, 4, 13, 3, 13, 4, 14, 2, 29, 2

Offset: 1

Rémy Sigrist, May 13 2020

nonn

This sequence has similarities with A088177; here we raise consecutive terms, there we multiply consecutive terms.

This sequence contains large runs of consecutive terms where every other term equals 2.

			The first terms, alongside a(n)^a(n+1), are:
  n   a(n)  a(n)^a(n+1)
  --  ----  -----------
   1     2            4
   2     2            8
   3     3            9
   4     2           16
   5     4           64
   6     3           27
   7     3           81
   8     4          256
   9     4         1024
  10     5           25
  11     2           32
  12     5          125
  13     3          243

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A334857

Cf. A088177, A334858.

PARI
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A338823 Lexicographically earliest sequence of positive integers such that for any distinct m and n, a(m) OR a(m+1) <> a(n) OR a(n+1) (where OR denotes the bitwise OR operator).

Original entry on oeis.org

1, 1, 2, 2, 4, 1, 6, 8, 1, 10, 2, 13, 1, 16, 2, 17, 4, 4, 8, 8, 16, 4, 18, 5, 24, 1, 26, 2, 28, 3, 32, 1, 36, 2, 32, 4, 24, 32, 7, 40, 1, 42, 2, 44, 1, 48, 2, 49, 4, 40, 8, 49, 6, 48, 4, 56, 2, 57, 4, 58, 5, 64, 1, 66, 2, 68, 3, 72, 1, 76, 2, 72, 4, 64, 8, 71

Offset: 1

Rémy Sigrist, Nov 11 2020

			The first terms, alongside a(n) OR a(n+1), are:
  n   a(n)  a(n) OR a(n+1)
  --  ----  --------------
   1     1               1
   2     1               3
   3     2               2
   4     2               6
   5     4               5
   6     1               7
   7     6              14
   8     8               9
   9     1              11
  10    10              10
  11     2              15
  12    13              13

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 10000 terms (where the color is function of the parity of n)
Rémy Sigrist, C program for A338823

Cf. A088177, A338824.

C
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Block[{a = {1, 1}, b = {1}}, Do[Block[{k = 1, m}, While[! FreeQ[b, Set[m, BitOr @@ {a[[-1]], k}]], k++]; AppendTo[a, k]; AppendTo[b, m]], {i, 3, 76}]; a] (* Michael De Vlieger, Nov 12 2020 *)

A349227 Lexicographically earliest sequence of positive integers such that the products of three consecutive terms are all distinct.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 1, 1, 5, 2, 2, 4, 3, 3, 1, 5, 5, 2, 3, 3, 3, 5, 4, 2, 4, 6, 3, 3, 7, 1, 1, 11, 2, 2, 7, 1, 5, 11, 2, 3, 7, 4, 2, 8, 5, 3, 5, 6, 5, 6, 7, 3, 5, 9, 5, 6, 8, 2, 7, 5, 4, 5, 8, 5, 7, 5, 7, 9, 3, 3, 11, 1, 7, 7, 2, 11, 4, 3, 9, 6, 4, 6, 7, 6, 7

Offset: 1

Rémy Sigrist, Nov 11 2021

nonn

This sequence has similarities with A088177; here we consider products of three consecutive terms, there products of two consecutive terms.

			The first terms, alongside a(n)*a(n+1)*a(n+2), are:
  n   a(n)  a(n)*a(n+1)*a(n+2)
  --  ----  ------------------
   1     1                   1
   2     1                   2
   3     1                   4
   4     2                   8
   5     2                  12
   6     2                   6
   7     3                   3
   8     1                   5
   9     1                  10
  10     5                  20

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

Cf. A088177, A349228.

s=0; pp=p=1; for (n=1, 86, for (v=1, oo, if (!bittest(s, q=pp*p*v), print1 (pp", "); s+=2^q; pp=p; p=v; break)))

def aupton(terms):
    alst, pset = [1, 1], set()
    for n in range(3, terms+1):
        p = p2 = alst[-1]*alst[-2]
        while p in pset: p += p2
        alst.append(p//p2); pset.add(p)
    return alst
print(aupton(86)) # Michael S. Branicky, Nov 12 2021

A349228 Products of three consecutive terms of A349227: a(n) = A349227(n) * A349227(n+1) * A349227(n+2).

Original entry on oeis.org

1, 2, 4, 8, 12, 6, 3, 5, 10, 20, 16, 24, 36, 9, 15, 25, 50, 30, 18, 27, 45, 60, 40, 32, 48, 72, 54, 63, 21, 7, 11, 22, 44, 28, 14, 35, 55, 110, 66, 42, 84, 56, 64, 80, 120, 75, 90, 150, 180, 210, 126, 105, 135, 225, 270, 240, 96, 112, 70, 140, 100, 160, 200

Offset: 1

Rémy Sigrist, Nov 11 2021

nonn

All terms are distinct.

Is this sequence a permutation of the natural numbers?

			a(5) = A349227(5) * A349227(6) * A349227(7) = 2 * 2 * 3 = 12.

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

Cf. A088177, A349227.

s=0; pp=p=1; for (n=1, 63, for (v=1, oo, if (!bittest(s, q=pp*p*v), print1 (q", "); s+=2^q; pp=p; p=v; break)))

def aupton(terms):
    A349227lst, plst, pset = [1, 1], [], set()
    for n in range(terms):
        p = p2 = A349227lst[-1]*A349227lst[-2]
        while p in pset: p += p2
        A349227lst.append(p//p2); plst.append(p); pset.add(p)
    return plst
print(aupton(63)) # Michael S. Branicky, Nov 12 2021

A354858 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and both the sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=2..n-1, and the product a(n) * a(n-1) is distinct from all previous products, a(i) * a(i-1), i=2..n-1.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 3, 9, 6, 8, 8, 10, 10, 12, 9, 15, 10, 16, 12, 15, 15, 18, 14, 20, 15, 21, 18, 20, 20, 22, 22, 24, 21, 27, 24, 26, 26, 28, 21, 35, 20, 38, 19, 57, 3, 63, 7, 56, 6, 58, 10, 55, 20, 52, 22, 55, 25, 60, 9, 69, 12, 70, 14, 72, 15, 75, 18, 70, 21, 75, 24, 68, 26, 72, 28, 74, 30, 65

Offset: 1

Scott R. Shannon, Jun 09 2022

This sequence uses a combination of the term selection rules of A354755 and A354753. The first forty-five terms are the same as A354755 beyond which they differ; see the examples below. In the first 500000 terms only six terms are prime, 2,3,7,19, with 2 and 3 occurring twice, the last being a(47) = 7. It is unknown if more appear. The only fixed points are 1,2,4,6, and it is likely no more exist.

			a(7) = 3 as a(6) = 6, and 3 is the smallest number that shares a factor with 6 and whose sum and product with the previous term, 6 + 3 = 9 and 6 * 3 = 18, have not previously appeared. Note 2 shares a factor with 6 but 6 + 2 = 8, and a sum of 8 has already occurred with a(4) + a(5) = 4 + 4 = 8, so 2 cannot be chosen.
a(46) = 63 as a(45) = 3, and 63 is the smallest number that shares a factor with 3 and whose sum and product with the previous term, 3 + 63 = 66 and 3 * 63 = 189, have not previously appeared. Note 60 shares a factor with 3 but the product 3 * 60 = 180 has already occurred with a(19) * a(20) = 12 * 15 = 180, so 60 cannot be chosen. This is the first term to differ from A354755.

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

Cf. A354755, A354753, A088177, A064413, A354803.

lista(nn) = my(va = vector(nn), vp = vector(nn-2), vs = vector(nn-2)); va[1] = 1; va[2] = 2; for (n=3, nn, my(k=2); while ((gcd(k, va[n-1]) == 1) || #select(x->(x==k*va[n-1]), vp) || #select(x->(x==k+va[n-1]), vs), k++); va[n] = k; vp[n-2] = k*va[n-1]; vs[n-2] = k+va[n-1];); va; \\ Michel Marcus, Jun 17 2022

cp's OEIS Frontend

A334857 Lexicographically earliest sequence of integers > 1 such that for any distinct m and n, a(m)^a(m+1) <> a(n)^a(n+1).

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A338823 Lexicographically earliest sequence of positive integers such that for any distinct m and n, a(m) OR a(m+1) <> a(n) OR a(n+1) (where OR denotes the bitwise OR operator).

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A349227 Lexicographically earliest sequence of positive integers such that the products of three consecutive terms are all distinct.

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A349228 Products of three consecutive terms of A349227: a(n) = A349227(n) * A349227(n+1) * A349227(n+2).

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A354858 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and both the sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=2..n-1, and the product a(n) * a(n-1) is distinct from all previous products, a(i) * a(i-1), i=2..n-1.

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PARI