A307720 -id:A307720 - cp's OEIS frontend

A307747 Terms in A307720 in order of appearance.

1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 13, 12, 14, 15, 17, 19, 16, 18, 23, 29, 20, 21, 24, 25, 26, 27, 22, 28, 30, 31, 37, 35, 33, 41, 43, 47, 53, 49, 39, 45, 32, 34, 36, 38, 59, 61, 55, 51, 57, 56, 58, 67, 71, 62, 63, 65, 64, 46, 48, 40, 44, 42, 50, 52, 54, 73

Offset: 1

Rémy Sigrist, Apr 26 2019

nonn

			The first terms of A307720 are (with periodic parts in parentheses):
    1,1,2,(1,3)*2,(2,2)*2,(2,3)*3,(3,3)*4,3,(4,2)*4,(4,3)*6,(5,1)*3,(7,1)*3,7,...
    *   *    *                               *               *       *
The terms marked with a star give the sequence:
    1,  2,   3,                              4,              5,      7,...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 10000 terms (red pixels correspond to prime terms)
Rémy Sigrist, C++ program for A307747

Cf. A307720.

A348459 a(n) = max(A307720(n), A307720(n+1)).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6

Offset: 1

N. J. A. Sloane, Oct 30 2021

nonn

This is the maximum of A348241 and A348242. It is the upper line (red or blue) in Cheswick's pictures in A348248.

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, PARI program for A348459

Cf. A307720, A307730, A348241, A348242, A348248, A348446.

PARI
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See Links section.
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A348244 Indices of 1's in A307720.

Original entry on oeis.org

1, 2, 4, 6, 48, 50, 52, 54, 56, 58, 1375, 1377, 1379, 1381, 1383, 1385, 1387, 1389, 1391, 1393, 1395, 1397, 3739, 3741, 3743, 3745, 3747, 3749, 3751, 3753, 3755, 3757, 3759, 3761, 3763, 3765, 3767, 3769, 3771, 3773, 6681, 6683, 6685, 6687, 6689, 6691, 6693, 6695, 6697, 6699, 6701, 6703, 6705

Offset: 1

N. J. A. Sloane, Oct 16 2021

nonn

Theorem: This sequence is infinite. Proof: Every prime in A307720 is immediately preceded or followed by a 1. - N. J. A. Sloane, Oct 30 2021

The start of this sequence consists of arithmetic progressions of constant difference 2 and lengths 1, 3, 6, 12, 18, 26, 76, ... that start with 1, 2, 48, 1375, 3739, 6681, 84627, 91833, ... It would be nice to have an alternative way to explain these numbers.

N. J. A. Sloane, Table of n, a(n) for n = 1..927

Cf. A307720.

A348458 Partial sums of A307720.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 12, 14, 16, 18, 20, 22, 25, 27, 30, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 66, 68, 72, 74, 78, 80, 84, 86, 90, 93, 97, 100, 104, 107, 111, 114, 118, 121, 125, 128, 133, 134, 139, 140, 145, 146, 153, 154, 161, 162, 169, 170, 177, 179, 184, 186, 191, 193, 198, 200, 205, 207

Offset: 1

N. J. A. Sloane, Oct 29 2021

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A307720, A307730.

A348581 a(n) is the least factor among all the products A307720(k) * A307720(k+1) equal to n.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Oct 24 2021

nonn

We know there are n ways to get n as a product of terms A307720(k)*A307720(k+1) for various k's. Look at these 2*n numbers from A307720. Then a(n) is the smallest of them.

			For n = 6:
- we have the following products equal to 6:
    A307720(7)  * A307720(8)  = 3 * 2 = 6
    A307720(12) * A307720(13) = 2 * 3 = 6
    A307720(13) * A307720(14) = 3 * 2 = 6
    A307720(14) * A307720(15) = 2 * 3 = 6
    A307720(15) * A307720(16) = 3 * 2 = 6
    A307720(16) * A307720(17) = 2 * 3 = 6
- the corresponding distinct factors are 2 and 3,
- hence a(6) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A348581

Cf. A307720, A307730, A348582.

C
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See Links section.
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a(p) = 1 for any prime number p.

a(n) * A348582(n) = n.

A348582 a(n) is the greatest factor among all the products A307720(k) * A307720(k+1) equal to n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 4, 13, 7, 5, 8, 17, 6, 19, 5, 7, 11, 23, 8, 5, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 7, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 8, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Oct 24 2021

nonn

We know there are n ways to get n as a product of terms A307720(k)*A307720(k+1) for various k's. Look at these 2*n numbers from A307720. Then a(n) is the largest of them.

			For n = 6:
- we have the following products equal to 6:
    A307720(7)  * A307720(8)  = 3 * 2 = 6
    A307720(12) * A307720(13) = 2 * 3 = 6
    A307720(13) * A307720(14) = 3 * 2 = 6
    A307720(14) * A307720(15) = 2 * 3 = 6
    A307720(15) * A307720(16) = 3 * 2 = 6
    A307720(16) * A307720(17) = 2 * 3 = 6
- the corresponding distinct factors are 2 and 3,
- hence a(6) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A348582

Cf. A307720, A307730, A348581.

C
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See Links section.
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a(p) = p for any prime number p.

a(n) * A348581(n) = n.

A307750 Records in A307720.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 23, 29, 30, 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

Offset: 1

Rémy Sigrist, Apr 26 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..5000
Rémy Sigrist, C++ program for A307750

See A307631 for the corresponding indices.

Cf. A307720.

A348460 a(n) = min(A307720(n), A307720(n+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

N. J. A. Sloane, Oct 30 2021

This is the minimum of A348241 and A348242. It is the lower line (red or blue) in Cheswick's pictures in A348248.

A companion to A348459.

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, PARI program for A348460

Cf. A307720, A307730, A348459.

PARI
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See Links section.
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A088177 a(1)=1, a(2)=1; for n>2, a(n) is the smallest positive integer such that the products a(i)*a(i+1), i=1..n-1, are all distinct.

Original entry on oeis.org

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

Offset: 1

John W. Layman, Sep 22 2003

A088178 is the sequence of distinct products a(i)a(i+1), i=1,2,3,... and appears to be a permutation of the natural numbers.
It appears that for k>2 the k-th occurrence of 1 lies between the first occurrences of primes p(2*k-4) and p(2*k-3).  For instance, the 5th occurrence of 1 lies between the first occurrences of 13 and 17, the 6th and 7th primes, respectively. - John W. Layman, Nov 16 2011
Note that a(n) = 1 for infinitely many n, because the sequence a(n) is not bounded and beside every new prime number must be the number 1. - Thomas Ordowski, Sep 04 2014. [This seems a rather sketchy argument, but I have a more complete proof using arguments similar to those we used in A098550. - N. J. A. Sloane, Oct 18 2021]
Example: ..., 5, 13, 1, 17, 2, 13, 3, 17, 4; ...
General: ..., k,  p, 1,  q, 2,  p, 3,  q, ..., k-1; ...
- Thomas Ordowski, Sep 08 2014

			Given that the sequence begins 1,1,2,2,... then a(5)=3, since either of the choices a(5)=1 or a(5)=2 would lead to a repetition of one of the previous products 1,2,4 of adjacent pairs of terms.

Michael De Vlieger, Table of n, a(n) for n = 1..10000, (first 1000 terms from T. D. Noe)

Cf. A088178, A307720, A307730, A341492, A348437, A348438, A348439.

Records: A348440, A348441.

A[1]:= 1: A[2]:= 1: S:= {1}:
for n from 3 to 100 do
  Sp:= select(type,map(s -> s/A[n-1],S),integer);
  if nops(Sp) = Sp[-1] then A[n]:= Sp[-1]+1
  else A[n]:= min({$1..Sp[-1]} minus Sp)
  fi;
  S:= S union {A[n-1]*A[n]};
od:
seq(A[n],n=1..100); # Robert Israel, Aug 28 2014

t = {1, 1}; Do[AppendTo[t, 1]; While[Length[Union[Most[t]*Rest[t]]] < n - 1, t[[-1]]++], {n, 3, 100}]; t (* T. D. Noe, Nov 16 2011 *)

from itertools import islice
def A088177(): # generator of terms
    yield 1
    yield 1
    p, a = {1}, 1
    while True:
        n = 1
        while n*a in p:
            n += 1
        p.add(n*a)
        a = n
        yield n
A088177_list = list(islice(A088177(),20)) # Chai Wah Wu, Oct 21 2021

a(n)*gcd(a(n-1),a(n+1)) = gcd(A088178(n-1),A088178(n)). - Thomas Ordowski, Jun 29 2015

A348246 Index of first occurrence of n in A307730.

Original entry on oeis.org

1, 2, 4, 8, 47, 7, 52, 27, 17, 60, 1374, 26, 1385, 59, 46, 137, 3738, 98, 3755, 204, 83, 1399, 6680, 136, 320, 1398, 176, 224, 6703, 345, 84626, 252, 1447, 3775, 375, 203, 84657, 3774, 1446, 502, 89480, 410, 89521, 1480, 319, 6733, 91832, 542, 452, 1024, 3847, 1618, 91879, 690, 1524, 501, 3846, 6732, 173092, 1074, 173151, 84695, 645, 807, 1579, 1735, 192882

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Oct 17 2021

nonn

a(n) is the least k such that A307730(k) = n.

			For n = 6:
- sequence A307730 starts: 1, 2, 2, 3, 3, 3, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6,
- the first occurrence of 6 appears at position 7,
- so a(6) = 7.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A348246

Cf. A307720, A307730, A307630, A348409.

Records: A348252, A348253.

C
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See Links section.
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A348409(n) - a(n) >= n - 1.

cp's OEIS Frontend

A307747 Terms in A307720 in order of appearance.

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A348459 a(n) = max(A307720(n), A307720(n+1)).

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PARI

A348244 Indices of 1's in A307720.

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A348458 Partial sums of A307720.

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A348581 a(n) is the least factor among all the products A307720(k) * A307720(k+1) equal to n.

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A348582 a(n) is the greatest factor among all the products A307720(k) * A307720(k+1) equal to n.

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C

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A307750 Records in A307720.

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A348460 a(n) = min(A307720(n), A307720(n+1)).

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PARI

A088177 a(1)=1, a(2)=1; for n>2, a(n) is the smallest positive integer such that the products a(i)*a(i+1), i=1..n-1, are all distinct.

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