A088177 -id:A088177 - cp's OEIS frontend

A088178 Sequence of distinct products b(n)*b(n+1), n=1,2,3,..., of the terms b(n) of A088177.

1, 2, 4, 6, 3, 5, 10, 8, 12, 9, 15, 20, 16, 24, 18, 21, 7, 11, 22, 14, 28, 32, 40, 25, 30, 36, 42, 35, 45, 27, 33, 44, 48, 60, 50, 70, 49, 56, 64, 72, 54, 66, 55, 65, 13, 17, 34, 26, 39, 51, 68, 52, 78, 84, 98, 63, 81, 90, 80, 88, 77, 91, 104, 96, 108, 99, 110, 100, 120, 132

Offset: 1

John W. Layman, Sep 22 2003

This is a permutation of the natural numbers (see the following comments).
Comments from Thomas Ordowski, Aug 24 2014 to Sep 07 2014: (Start)
If a(n) is a prime then a(m) > a(n) for m > n.
Conjecture: the term a(n) is a prime if and only if every number < a(n) belongs to the set {a(1), a(2), ..., a(n-1)}.
The numbers in A033476 appear in increasing order.
It seems that the squarethe terms in s of the natural numbers also appear in increasing order, but A087811 are not strictly increasing.
Lemma: the sequence a(n) is a permutation of all natural numbers iff b(n) = 1 for infinitely many n, where b(n) = A088177(n), because after every b(n) = 1 is the smallest missing number in the sequence a(n).
Theorem: the sequence a(n) is a permutation of the natural numbers. Proof: see my note to A088177.
At most two consecutive terms can form a decreasing subsequence.
(End)
An equivalent definition. At step n, choose a(n) to be the smallest unused multiple of the auxiliary number r, which is initially 1 and is changed to a(n)/r after each step. - Ivan Neretin, May 04 2015
Considered as a permutation of the positive integers, there are finite cycles (1), (2), (3, 4, 6, 5), (8), (11, 18, 15), (52), and probably others. The cycle containing 7, on the other hand, is ( ..., 85, 46, 17, 7, 10, 9, 12, 20, 14, 24, 25, 30, 27, 42, 66, 99, 160, 308, 343, 430, 517, 902, ... ), and may be infinite. The inverse permutation is A341492. - N. J. A. Sloane, Oct 19 2021

Ivan Neretin, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michael De Vlieger)

Cf. A088177, A341492, A348437-A348439.

Records: A348442, A348443.

a088177[n_Integer] := Module[{t = {1, 1}}, Do[AppendTo[t, 1]; While[Length[Union[Most[t]*Rest[t]]] < i - 1, t[[-1]]++], {i, 3, n}]; t]; a088178[n_Integer] := Last[a088177[n]]*Last[a088177[n + 1]]; a088178 /@ Range[120] (* Michael De Vlieger, Aug 30 2014, based on T. D. Noe's script at A088177 *)

from itertools import islice
def A088178(): # generator of terms
    yield 1
    p, a = {1}, 1
    while True:
        n, na = 1, a
        while na in p:
            n += 1
            na += a
        p.add(na)
        a = n
        yield na
A088178_list = list(islice(A088178(),20)) # Chai Wah Wu, Oct 21 2021

a(n) = A088177(n)* A088177(n+1).

a(m) < a(n)^2 for m < n. - Thomas Ordowski, Sep 02 2014

Edited by N. J. A. Sloane, Oct 18 2021

A348440 Records in A088177.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 12, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

Apparently the terms agree with the prime numbers A000040, beginning at 29. - Hugo Pfoertner, Oct 21 2021

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A088177, A088178, A348441.

Block[{c, m = 1, n}, Union@ FoldList[Max, {1}~Join~Reap[Do[n = 1; While[IntegerQ[c[m n]], n++]; Sow[n]; Set[c[m n], 1]; m = n, 2^12]][[-1, -1]]]] (* Michael De Vlieger, Oct 21 2021 *)

from itertools import islice
def A348440(): # 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 < n:
            c = n
            yield c
A348440_list = list(islice(A348440(),100)) # Chai Wah Wu, Oct 21 2021

A348438 Where prime(n) appears for the first time in A088177.

Original entry on oeis.org

3, 5, 7, 17, 19, 45, 47, 83, 85, 169, 171, 181, 183, 193, 195, 205, 207, 533, 535, 561, 563, 585, 587, 784, 786, 1040, 1042, 1068, 1070, 1096, 1098, 1126, 1128, 1150, 1152, 1932, 1934, 1986, 1988, 2022, 2024, 2062, 2064, 2090, 2092, 2118, 2120, 2146, 2148, 2178, 2180, 2206, 2208, 3929, 3931, 3965, 3967

Offset: 1

N. J. A. Sloane, Oct 19 2021

nonn

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

Cf. A088177, A088178, A348439.

A348437 Where n appears for the first time in A088177.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 17, 23, 30, 36, 19, 34, 45, 55, 79, 77, 47, 111, 83, 99, 93, 143, 85, 247, 165, 231, 229, 239, 169, 329, 171, 353, 333, 417, 227, 503, 181, 465, 349, 457, 183, 463, 193, 686, 515, 203, 195, 856, 309, 848, 623, 217, 205, 988, 553, 920, 449, 213, 207, 1012, 533, 840, 842, 896

Offset: 1

N. J. A. Sloane, Oct 18 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..500 (terms 1..189 from N. J. A. Sloane)

Cf. A088177, A088178, A341492, A348438.

A348441 Indices of records in A088177.

Original entry on oeis.org

1, 3, 5, 7, 15, 17, 19, 34, 45, 47, 83, 85, 165, 169, 171, 181, 183, 193, 195, 205, 207, 533, 535, 561, 563, 585, 587, 784, 786, 1040, 1042, 1068, 1070, 1096, 1098, 1126, 1128, 1150, 1152, 1932, 1934, 1986, 1988, 2022, 2024, 2062, 2064, 2090, 2092, 2118, 2120, 2146, 2148, 2178, 2180, 2206, 2208

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..381

Cf. A088177, A088178, A348440.

Block[{a, c, m = 1, n}, a = {1}~Join~Reap[Do[n = 1; While[IntegerQ[c[m n]], n++]; Sow[n]; Set[c[m n], i]; m = n, {i, 3000}]][[-1, -1]]; Map[FirstPosition[a, #][[1]] &, Union@ FoldList[Max, a]]] (* Michael De Vlieger, Oct 21 2021 *)

from itertools import islice
def A348441(): # generator of terms
    yield 1
    c, p, a, i = 1, {1}, 1, 2
    while True:
        n, na = 1, a
        while na in p:
            n += 1
            na += a
        p.add(na)
        a = n
        i += 1
        if c < n:
            c = n
            yield i
A348441_list = list(islice(A348441(),100)) # Chai Wah Wu, Oct 21 2021

A341490 Numbers k such that A088177(k) = 1.

Original entry on oeis.org

1, 2, 6, 18, 46, 84, 170, 182, 194, 206, 534, 562, 586, 785, 1041, 1069, 1097, 1127, 1151, 1933, 1987, 2023, 2063, 2091, 2119, 2147, 2179, 2207, 3930, 3966, 4002, 4038, 4074, 4110, 4220, 6012, 8503, 8543, 8583, 8623, 8815, 8877, 8947, 9013, 9061, 9109, 9157

Offset: 1

Rémy Sigrist, Feb 13 2021

nonn

This sequence is infinite, and A088178 is a permutation of the positive integers.

			A088177(6) = 1, so 6 belongs to this sequence.
A088177(7) = 5, so 7 does not belong to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..5000
Rémy Sigrist, C++ program for A341490

Cf. A088177, A088178 .

A354753 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 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, 2, 10, 4, 8, 6, 3, 3, 9, 6, 6, 10, 5, 5, 15, 3, 21, 6, 12, 8, 8, 10, 10, 12, 9, 9, 15, 6, 14, 2, 22, 4, 14, 7, 7, 21, 9, 18, 8, 14, 10, 15, 12, 14, 14, 16, 8, 20, 10, 22, 6, 26, 2, 34, 4, 26, 8, 22, 11, 11, 33, 3, 39, 6, 32, 8, 30, 9, 24, 12, 21, 14, 20, 15, 15, 21, 18, 18

Offset: 1

Scott R. Shannon, Jun 06 2022

This sequence uses a similar rule to A088177 but here all neighboring terms also share a factor. In the first 500000 terms the fixed points are 1,2,4,6, it is likely no more exist, while the smallest number not to have appeared is 1153. The sequence is conjectured to be a permutation of the positive integers.

See A354754 for the products of all pairs of terms.

			a(7) = 2 as a(6) = 6 and 2 is the smallest positive number that shares a factor with 6 and whose product with 6, 2 * 6 = 12, has not previously appeared.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n) n = 1..2^14, showing records in red, a(n) = 2 in blue, fixed points highlighted in gold.
Scott R. Shannon, Image of the first 50000 terms.

Cf. A354754, A354803, A354804, A354749, A354759, A088177, A064413.

nn = 120; c[] = 0; a[1] = c[1] = 1; a[2] = a[2] = j = 2; Do[k = 2; While[Nand[c[j*k] == 0, ! CoprimeQ[j, k]], k++]; Set[{a[n], c[j*k]}, {k, n}]; j = k, {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jun 15 2022 *)

lista(nn) = my(va = vector(nn), vp = 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), k++); va[n] = k; vp[n-2] = k*va[n-1];); va; \\ Michel Marcus, Jun 14 2022

A354803 a(1) = 1; for n > 1, a(n) is the smallest positive number that is coprime to a(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, 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, 37, 1, 32, 3, 16, 7, 12, 11, 13, 6, 17, 3, 19, 4, 17, 5, 19, 6, 23, 3, 25, 4, 23

Offset: 1

Scott R. Shannon, Jun 07 2022

This sequences uses similar a similar rule to A088177 but here all neighboring terms are coprime. In the first 1000000 terms the only fixed point is the first term while the smallest number not to have appeared is 2054. The sequence is conjectured to be a permutation of the positive integers.

See A354804 for the products of all pairs of terms.

			a(5) = 1 as a(4) = 3 and 1 is the smallest positive number that is coprime to 3 and whose product with 3, 1 * 3 = 3, has not previously appeared.

Scott R. Shannon, Image of the first 500000 terms.

Cf. A354804, A354749, A354759, A354753, A354754, A088177, A317024.

A354749 a(1) = 1; for n > 1, a(n) is the smallest positive number greater than 1 that is coprime to a(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, 3, 4, 5, 2, 7, 3, 5, 6, 7, 4, 9, 2, 11, 3, 8, 5, 7, 8, 9, 5, 11, 4, 13, 2, 17, 3, 13, 5, 12, 7, 9, 10, 7, 11, 6, 13, 7, 15, 8, 11, 9, 13, 8, 17, 4, 19, 2, 23, 3, 16, 5, 17, 6, 19, 3, 25, 2, 27, 4, 23, 5, 19, 7, 16, 9, 14, 11, 10, 13, 11, 12, 13, 14, 15, 11, 16, 13, 15, 16, 17, 7, 20, 9

Offset: 1

Scott R. Shannon, Jun 06 2022

This is a variation of A354803 where all terms beyond the first must be greater than 1. In the first 1000000 terms the fixed points are 1,2,3,4,5,7, it is likely no more exist, while the smallest number not to have appeared is 2090. The sequence is conjectured to be a permutation of the positive integers.

See A354759 for the products of all pairs of terms.

			a(6) = 2 as a(5) = 5 and 2 is the smallest number greater than 1 that is coprime to 5 and whose product with 5, 2 * 5 = 10, has not previously appeared.

Scott R. Shannon, Image of the first 500000 terms.

Cf. A354759, A354803, A354804, A354753, A354754, A088177.

A354754 The products of consecutive terms in A354753.

Original entry on oeis.org

2, 4, 8, 16, 24, 12, 20, 40, 32, 48, 18, 9, 27, 54, 36, 60, 50, 25, 75, 45, 63, 126, 72, 96, 64, 80, 100, 120, 108, 81, 135, 90, 84, 28, 44, 88, 56, 98, 49, 147, 189, 162, 144, 112, 140, 150, 180, 168, 196, 224, 128, 160, 200, 220, 132, 156, 52, 68, 136, 104, 208, 176, 242, 121, 363, 99

Offset: 1

Scott R. Shannon, Jun 06 2022

nonn

See A354753 for further details.

			a(6) = 12 as A354753(6) * A354753(7) = 6 * 2 = 12.

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

Cf. A354753, A354803, A354804, A354749, A354759, A088177, A064413.

cp's OEIS Frontend

A088178 Sequence of distinct products b(n)*b(n+1), n=1,2,3,..., of the terms b(n) of A088177.

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A348440 Records in A088177.

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A348438 Where prime(n) appears for the first time in A088177.

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A348437 Where n appears for the first time in A088177.

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A348441 Indices of records in A088177.

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A341490 Numbers k such that A088177(k) = 1.

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A354753 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 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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A354803 a(1) = 1; for n > 1, a(n) is the smallest positive number that is coprime to a(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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A354749 a(1) = 1; for n > 1, a(n) is the smallest positive number greater than 1 that is coprime to a(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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A354754 The products of consecutive terms in A354753.