A307720 -id:A307720 - cp's OEIS frontend

A307730 a(n) = A307720(n) * A307720(n+1).

1, 2, 2, 3, 3, 3, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 9, 9, 9, 9, 12, 8, 8, 8, 8, 8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 15, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 14, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 14, 14, 14, 14, 14, 14, 14

Offset: 1

Rémy Sigrist, Apr 25 2019

For all positive integers n, n appears n times.

			The first terms in this sequence and in A307720 are:
  n   a(n)  A307720(n)
  --  ----  ----------
   1     1           1
   2     2           1
   3     2           2
   4     3           1
   5     3           3
   6     3           1
   7     6           3
   8     4           2
   9     4           2
  10     4           2

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, Scatterplot of the first 10000000 terms
Rémy Sigrist, PARI program for A307730
N. J. A. Sloane, Table of n, a(n) for n = 1..999999

Cf. A002024, A088178, A307720.

Cf. A348579 (indices of occurrence of each number), A348246 (first occurrence of each number), A348409 (last occurrence).

PARI
```
\\ See Links section.
```

from itertools import islice
from collections import Counter
def A307730(): # generator of terms. Greedy algorithm
    c, b = Counter(), 1
    while True:
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        c[kb] += 1
        b = k
        yield kb
A307730_list = list(islice(A307730(),100)) # Chai Wah Wu, Oct 21 2021

A307632 Index of first occurrence of n-th prime in A307720.

Original entry on oeis.org

3, 5, 47, 53, 1374, 1386, 3738, 3756, 6680, 6704, 84626, 84658, 89480, 89522, 91832, 91880, 173092, 173152, 192882, 192950, 524587, 524661, 865301, 865385, 876543, 876641, 890479, 890583, 904273, 904383, 918859, 918987, 1628979, 1629117, 1647107, 1647257, 1666775

Offset: 1

N. J. A. Sloane, Apr 26 2019

nonn

It follows from the definition of A307720 that if p = k-th prime, k>1 and k odd, and q = (k+1)st prime, then the first time p appears in the sequence is at the start of a subsequence (p,1) [(p-1)/2 times], p, (1,q) [(q+1)/2 times].
For example, the fifth prime (11) first appears in A307720 at step 1374 at the start of the subsequence 11, 1, 11, 1, 11, 1, 11, 1, 11, 1, 11, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13.
So q appears p+1 steps after p, which explains why the terms of the present sequence appear in pairs.
In fact, it appears that one can make a much stronger statement about what happens immediately after the first occurrence of p. Look at the terms in A307720 following the first 11 at step 1374. It may be that the next O(p^2) terms are forced.

N. J. A. Sloane, Table of n, a(n) for n = 1..1229 (taken from Rémy Sigrist's b-file for A307630; first 132 terms from Hans Havermann)

Cf. A307720, A307630, A307631.

First differences = A348773.

More terms from Hans Havermann, Apr 26 2019

A348248 Let d = A307720(n) - A307720(n-1); a(n) = 0 if d = 0; a(n) = 1 if d > 0; a(n) = 2 if d < 0.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 0, 0, 0, 0, 1, 2, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 2

N. J. A. Sloane, Oct 20 2021

nonn

If one looks at the graph of A307720 (that entry has a number of versions of this graph besides the one that appears when the "graph" button is clicked), one sees that initially A307720(n) is usually greater than A307720(n-1) if n is odd.
Think of A307720 as a piano score in which normally the right hand (n = 2*i+1) is to the right of the left hand (n = 2*i).
However, as can be seen in William Cheswick's colored plots, sometimes the right and left hands swap. In these plots, the right-hand points (n odd) are blue and the left-points (n even) are red.
A run of terms 121212... in the present sequence is a normal sequence of notes left, right, left, right, ..., where blue is on top.
A run 212121... indicates that the hands have been swapped (red is on top).
A run 000000... indicates that both hands are playing the same note (red and blue are at the same level).
The purpose of the present sequence and related sequences is to study when the hands swap. At present there is no explanation for when this happens.
The sequence of pictures suggests that there will be infinitely many occasions when the hands swap. The upper color in the picture will alternate infinitely often between red and blue, with longer and longer runs before the upper color changes.

William Cheswick, Colored plot of first 200 terms of A307720 (See Comments for explanation of colors in these pictures)
William Cheswick, Colored plot of first 1000 terms of A307720
William Cheswick, Colored plot of first 10000 terms of A307720
William Cheswick, Colored plot of first 10^5 terms of A307720
William Cheswick, Colored plot of first 10^6 terms of A307720
Rémy Sigrist, Scatterplot of the first 10000000 terms of A307720
Chai Wah Wu, Scatterplot of the first 100 million terms of A348446 [Shows how the lead changes between the left and right hands. The color scheme here is different from that in _William Cheswick_'s pictures.]

Cf. A307720, A307730, A348446, A348459.

A307630 Index at which n first appears in A307720.

Original entry on oeis.org

1, 3, 5, 27, 47, 99, 53, 137, 177, 1024, 1374, 2474, 1386, 3326, 3662, 5274, 3738, 6290, 3756, 8954, 9374, 12878, 6680, 9682, 9850, 10324, 11010, 14578, 6704, 78506, 84626, 106968, 88474, 127682, 86802, 143544, 84658, 160664, 97850, 274079, 89480, 326195

Offset: 1

N. J. A. Sloane, Apr 25 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (terms 1..239 from Hans Havermann, terms 240..669 from Lars Blomberg)
Rémy Sigrist, Log-plot of first 5000 terms.
Rémy Sigrist, C++ program for A307630.

Cf. A307720, A307631, A307632, A307633, A307634, A307730, A307747.

More terms from Hans Havermann, Apr 25 2019

A307631 Indices of record high-points in A307720.

Original entry on oeis.org

1, 3, 5, 27, 47, 53, 137, 177, 1024, 1374, 1386, 3326, 3662, 3738, 3756, 6680, 6704, 78506, 84626, 84658, 89480, 89522, 91832, 91880, 173092, 173152, 192882, 192950, 524587, 524661, 865301, 865385, 876543, 876641, 890479, 890583, 904273, 904383, 918859, 918987

Offset: 1

N. J. A. Sloane, Apr 25 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..5000 (first 266 terms from Lars Blomberg)
Rémy Sigrist, PARI program for A307631
Rémy Sigrist, C++ program for A307631

Cf. A307720, A307630.

See Links section.
(C++) See Links section.

More terms from Rémy Sigrist, Apr 25 2019

A348241 A307720(2*n+1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 15 2021

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..14222

Cf. A307720, A348242.

A348242 A307720(2*n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 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, 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, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4

Offset: 1

N. J. A. Sloane, Oct 15 2021

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..14223

Cf. A307720, A348241.

A307633 RUNS transform of bisection {A307720(2k+1): k>=0}.

Original entry on oeis.org

1, 1, 2, 2, 7, 10, 3, 4, 5, 7, 7, 9, 10, 20, 14, 10, 14, 16, 18, 45, 21, 25, 20, 24, 28, 22, 27, 31, 68, 41, 25, 30, 35, 40, 45, 12, 24, 17, 22, 28, 19, 26, 32, 72, 39, 44, 49, 105, 61, 42, 48, 54, 60, 66, 45, 52, 59, 49, 56, 63, 38, 18, 36, 26, 34, 42, 51, 53, 67, 210, 231, 64, 72, 80, 88

Offset: 1

N. J. A. Sloane, Apr 27 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A307633

Cf. A307720, A307634.

PARI
```
See Links section.
```

A307634 RUNS transform of bisection {A307720(2k): k>=1}.

Original entry on oeis.org

3, 5, 5, 4, 6, 6, 12, 27, 8, 25, 58, 13, 15, 63, 72, 81, 32, 76, 175, 6, 7, 11, 13, 66, 78, 33, 39, 132, 50, 115, 270, 156, 168, 38, 9, 10, 17, 19, 153, 120, 65, 70, 75, 71, 77, 82, 304, 204, 81, 114, 12, 15, 23, 29, 138, 57, 69, 228, 85, 90, 95, 93, 99, 104, 210, 74, 80, 84, 63, 174, 78, 87

Offset: 1

N. J. A. Sloane, Apr 27 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A307634

Cf. A307720, A307633.

PARI
```
See Links section.
```

A348446 a(n) = A307720(2n-1) - A307220(2n).

Original entry on oeis.org

0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 4, 4, 4, 6, 6, 6, 5, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 4, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Oct 22 2021

sign

My guess is that this changes signs infinitely often, but is more likely to be positive than negative. Perhaps the behavior is akin to that of A066520, which shows the "great prime race" between primes congruent to 3 mod 4 and primes congruent to 1 mod 4.

See also the graphs in A307720 and A348248.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Rémy Sigrist, Scatterplot of the first million terms
Rémy Sigrist, Scatterplot of the first five million terms
N. J. A. Sloane, Table of n, a(n) for n = 1..500000
Chai Wah Wu, Scatterplot of the first 100 million terms

Cf. A066520, A307720, A307730, A348248.

from itertools import islice
from collections import Counter
def A348446(): # generator of terms. Greedy algorithm
    a = 1
    c, b = Counter(), 1
    while True:
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        c[kb] += 1
        b = k
        a2 = k
        yield a-a2
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        c[kb] += 1
        b = k
        a = k
A348446_list = list(islice(A348446(),100)) # Chai Wah Wu, Oct 23 2021

cp's OEIS Frontend

A307730 a(n) = A307720(n) * A307720(n+1).

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A307632 Index of first occurrence of n-th prime in A307720.

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A348248 Let d = A307720(n) - A307720(n-1); a(n) = 0 if d = 0; a(n) = 1 if d > 0; a(n) = 2 if d < 0.

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A307630 Index at which n first appears in A307720.

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A307631 Indices of record high-points in A307720.

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A348241 A307720(2*n+1).

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A348242 A307720(2*n).

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A307633 RUNS transform of bisection {A307720(2k+1): k>=0}.

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PARI

A307634 RUNS transform of bisection {A307720(2k): k>=1}.

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PARI

A348446 a(n) = A307720(2*n-1) - A307220(2*n).

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Programs

Python

A348446 a(n) = A307720(2n-1) - A307220(2n).