A307720 -id:A307720 - cp's OEIS frontend

A348409 a(n) is the greatest k such that A307730(k) = n.

1, 3, 6, 11, 51, 16, 58, 34, 25, 69, 1384, 45, 1397, 82, 97, 152, 3754, 115, 3773, 223, 135, 1420, 6702, 175, 344, 1445, 202, 251, 6731, 374, 84656, 283, 1479, 3808, 409, 318, 84693, 3845, 1617, 541, 89520, 451, 89563, 1523, 689, 6778, 91878, 589, 500, 1073

Offset: 1

Rémy Sigrist, Oct 17 2021

nonn

			For n = 6:
- sequence A307730 starts: 1, 2, 2, 3, 3, 3, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6,
- the sixth occurrence of 6 appears at position 16,
- so a(6) = 16.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A348409

Cf. A307720, A307730, A348246.

Records: A348253, A348254.

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a(n) - A348246(n) >= n - 1.

A348249 a(n) = A348409(n) - A348246(n) + 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 7, 8, 9, 10, 11, 20, 13, 24, 52, 16, 17, 18, 19, 20, 53, 22, 23, 40, 25, 48, 27, 28, 29, 30, 31, 32, 33, 34, 35, 116, 37, 72, 172, 40, 41, 42, 43, 44, 371, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 144, 2662, 104, 59, 60, 61, 62, 161, 64, 155, 66, 67, 68, 69, 70, 71, 136, 73, 136

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Oct 20 2021

nonn

This is the length of time from when A307720 produces the first n to when it produces the final (n-th) n. Cf. A307730. By definition, a(n) >= n.

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

Cf. A307720, A307730, A348245, A348246, A348409.

Records: A348255, A348256.

A348252 Records in A348246.

Original entry on oeis.org

1, 2, 4, 8, 47, 52, 60, 1374, 1385, 3738, 3755, 6680, 6703, 84626, 84657, 89480, 89521, 91832, 91879, 173092, 173151, 192882, 192949, 524587, 524660, 865301, 865384, 876543, 876640, 890479, 890582, 904273, 904382, 918859, 918986, 1628979, 1629116, 1647107, 1647256, 1666775, 1666932, 1687763, 1687930

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..7053
Rémy Sigrist, C program for A348252

Cf. A307720, A307730, A348246, A348409, A348249, A348253-A348256.

C
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// See Links section.
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A348256 Indices of records in A348249.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 12, 14, 15, 21, 36, 39, 45, 57, 87, 203, 319, 429, 435, 517, 530, 710, 790, 1310, 1730, 2290, 3370, 3830, 4210, 7430, 7570, 9530, 9710, 13010, 44810, 70270

Offset: 1

N. J. A. Sloane, Oct 21 2021

Rémy Sigrist, C program for A348256

Cf. A307720, A307730, A348246, A348409, A348249, A348252-A348255.

C
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a(35)-a(37) from Rémy Sigrist, Oct 22 2021

A348243 Successive terms in A307730 but listing just one of each run of identical terms.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 16 2021

nonn

Obviously the graph here is a compressed version of the graph of A307730.

			A307730 begins 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, ... Replacing runs of identical terms by a single number, we get 1, 2, 3, 6, 4, 6, 9, 12, 8, ...

Rémy Sigrist, Table of n, a(n) for n = 1..25000 (first 261 terms from N. J. A. Sloane)
Rémy Sigrist, C program for A348243

Cf. A307720, A307730, A348244-A348248.

C
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See Links section.
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A348773 First differences of A307632.

Original entry on oeis.org

2, 42, 6, 1321, 12, 2352, 18, 2924, 24, 77922, 32, 4822, 42, 2310, 48, 81212, 60, 19730, 68, 331637, 74, 340640, 84, 11158, 98, 13838, 104, 13690, 110, 14476, 128, 709992, 138, 17990, 150, 19518, 158, 20830, 168, 2277394, 180, 62350, 192, 82484, 198, 76962, 212, 84852, 228, 15407670, 234, 87388, 242, 90636

Offset: 1

N. J. A. Sloane, Nov 06 2021

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..1228

Cf. A307720, A307730, A307632.

Bisections: A348774, A348775.

A307707 Lexicographically earliest sequence of nonnegative integers in which, for all k >= 0, there are exactly k pairs of consecutive terms whose sum is k.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Apr 23 2019

The old definition was "Lexicographically earliest sequence starting with a(1) = 0 such that a(n) is the number of pairs of contiguous terms whose sum is a(n)".
From Paul Curtz, Apr 27 2019: This can be written as a triangle:
                    0
                  1   1
                1   2   1
              2   2   2   2
            2   3   2   3   2
          3   3   3   3   3   3
        3   4   3   4   3   4   3
...

Jean-Marc Falcoz, Table of n, a(n) for n = 1..11326

Cf. A002024.
Cf. also A007590, A057353, A106466 and A238410.
For other versions see A307720 and A378117.

m = 107; a[1]=0;
a24[n_] := Ceiling[(Sqrt[8n+1]-1)/2];
Array[a, m] /. Solve[Table[a[n] + a[n+1] == a24[n], {n, 1, m-1}]][[1]] (* Jean-François Alcover, Jun 02 2019, after Rémy Sigrist's formula *)

v=0; rem=wanted=1; for (n=1, 107, print1 (v", "); v=wanted-v; if (rem--==0, rem=wanted++)) \\ Rémy Sigrist, Apr 23 2019

a(n) + a(n+1) = A002024(n). - Rémy Sigrist, Apr 24 2019

Let t_m = m*(m+1)/2. Write n = t_m - i with m >= 1 and 0 <= i < m. Then a(n) = m/2 if m is even, or if m is odd, a(n) = (m-1)/2 + (i-1 mod 2). - N. J. A. Sloane, Nov 16 2024

Definition clarified by Rémy Sigrist and N. J. A. Sloane, Nov 17 2024

A348245 If A307730(n) = i, then a(n) = 1 + number of copies of i that have already appeared in A307730.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 16 2021

nonn

From the definition of A307720 and A307730, we know that any number i must eventually appear exactly i times in A307730. So a(n) <= A307730(n).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Table of n, a(n) for n = 1..999999

Cf. A307720, A307730, A348243.

A348444 is another version.

A348253 Indices of records in A348246.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 11, 13, 17, 19, 23, 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

Cf. A307720, A307730, A348246, A348409, A348249, A348252-A348256.

A348774 A348773(2*n+1).

Original entry on oeis.org

2, 6, 12, 18, 24, 32, 42, 48, 60, 68, 74, 84, 98, 104, 110, 128, 138, 150, 158, 168, 180, 192, 198, 212, 228, 234, 242, 258, 270, 278, 284, 308, 314, 332, 348, 354, 368, 380, 390, 402, 420, 432, 440, 450, 462, 468, 488, 500, 510, 524, 548, 564, 572, 588, 600, 608, 618, 632, 644, 654

Offset: 0

N. J. A. Sloane, Nov 07 2021

nonn

The first differences are 4, 6, 6, 6, 8, ... and apart from the initial term4, appears to coincide with A155067, the differences between successive odd-indexed primes. If confirmed, this will be one of the few formulas known for A307720.

The other bisection of A348773, A348775, seems much more mysterious.

N. J. A. Sloane, Table of n, a(n) for n = 0..613

Cf. A155067, A307720, A307730, A307632, A348773, A348775.

cp's OEIS Frontend

A348409 a(n) is the greatest k such that A307730(k) = n.

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A348249 a(n) = A348409(n) - A348246(n) + 1.

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A348252 Records in A348246.

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C

A348256 Indices of records in A348249.

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C

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A348243 Successive terms in A307730 but listing just one of each run of identical terms.

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C

A348773 First differences of A307632.

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A307707 Lexicographically earliest sequence of nonnegative integers in which, for all k >= 0, there are exactly k pairs of consecutive terms whose sum is k.

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Mathematica

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A348245 If A307730(n) = i, then a(n) = 1 + number of copies of i that have already appeared in A307730.

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A348253 Indices of records in A348246.

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A348774 A348773(2*n+1).

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