A307730 -id:A307730 - cp's OEIS frontend

A348246 Index of first occurrence of n in A307730.

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.

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

Original entry on oeis.org

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.

C
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See Links section.
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a(n) - A348246(n) >= n - 1.

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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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.

A348449 Indices of records in A307730.

Original entry on oeis.org

1, 2, 4, 7, 17, 26, 46, 83, 136, 176, 203, 319, 452, 501, 645, 806, 942, 1023, 1373, 2142, 2352, 2473, 3013, 4392, 4812, 5273, 5881, 7820, 8360, 8953, 9373, 11998, 12646, 13371, 13875, 15944, 17294, 18152, 20084, 24447, 25143, 26521, 27564, 29090, 33764, 40156, 41482, 42826, 44217, 46647, 51069

Offset: 1

N. J. A. Sloane, Oct 23 2021

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 110 terms from N. J. A. Sloane)
Rémy Sigrist, C program for A348449

C
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// See Links section.
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A348247 Where prime(n) first appears in A307730.

Original entry on oeis.org

2, 4, 47, 52, 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

Offset: 1

N. J. A. Sloane, Oct 17 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..134

Cf. A307720, A307730, A307632, A348246.

from collections import Counter
from sympy import prime
def A348247(n):
    c, b, p, i = Counter(), 1, prime(n), 1
    while True:
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        if kb == p:
            return i
        c[kb] += 1
        b = k
        i += 1 # Chai Wah Wu, Oct 21 2021

More terms from Chai Wah Wu, Oct 21 2021

A348251 Numbers that take more than the minimum number of steps to appear in A307730.

Original entry on oeis.org

6, 12, 14, 15, 21, 24, 26, 36, 38, 39, 45, 56, 57, 58, 63, 65, 72, 74, 78, 85, 86, 87, 90, 91, 99, 106, 110, 111, 115, 119, 122, 126, 129, 132, 135, 138, 142, 143, 145, 150, 158, 159, 161, 165, 171, 174, 175, 176, 178, 183, 185, 187, 190, 202, 203, 209, 213, 214, 215, 216, 220, 222, 226, 230, 231

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

The total number of appearances of a number n in A307730 is required to be n. For the numbers listed here, those appearances are spread out over more than n terms of A307730.
These are the numbers n for which A348250(n) is nonzero.
It would be nice to have an independent characterization of these numbers. It is a theorem that no primes can appear, but what more can be said? The graph is roughly linear, and the first differences are small but irregular.

N. J. A. Sloane, Table of n, a(n) for n = 1..3642

Cf. A307720, A307730, A348246, A348249, A348409, A348250.

A348448 Records in A307730.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 15, 21, 24, 27, 36, 45, 49, 56, 63, 72, 81, 90, 99, 110, 121, 132, 143, 150, 165, 176, 187, 190, 209, 220, 231, 243, 253, 264, 275, 290, 319, 324, 351, 360, 375, 377, 406, 435, 459, 475, 476, 493, 522, 551, 567, 575, 588, 609, 638, 667, 675, 700, 725, 754, 783, 810, 837, 858

Offset: 1

N. J. A. Sloane, Oct 23 2021

nonn

The points with coordinates (x,y) = (A348449(n), A348448(n)), n = 1,2,3,... are the upper envelope of the graph of A307730. Based on the first 10^6 terms of A307730 we get 110 points on the upper envelope, which appear to lie very roughly on the curve y = x^0.57. It would be nice to have a more trustworthy expression for this curve!

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 110 terms from N. J. A. Sloane)
Rémy Sigrist, Scatterplot of the first 10000000 terms of A307730
Rémy Sigrist, C program for A348448
N. J. A. Sloane, Table of n, A307730(n) for n = 1..999999

Cf. A307720, A307730, A348247, A348449.

C
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// See Links section.
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A348579 Triangle T(n, k), n > 0, 0 < k <= n, read by rows; the n-th row contains, in ascending order, the numbers m such that A307730(m) = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 47, 48, 49, 50, 51, 7, 12, 13, 14, 15, 16, 52, 53, 54, 55, 56, 57, 58, 27, 28, 29, 30, 31, 32, 33, 34, 17, 18, 19, 20, 21, 22, 23, 24, 25, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 1374, 1375, 1376, 1377, 1378, 1379, 1380, 1381, 1382, 1383, 1384

Offset: 1

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

			Triangle T(n, k) begins:
  n\k|   1   2   3   4   5   6   7   8   9  10
  ---+----------------------------------------
    1|   1
    2|   2   3
    3|   4   5   6
    4|   8   9  10  11
    5|  47  48  49  50  51
    6|   7  12  13  14  15  16
    7|  52  53  54  55  56  57  58
    8|  27  28  29  30  31  32  33  34
    9|  17  18  19  20  21  22  23  24  25
   10|  60  61  62  63  64  65  66  67  68  69

Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141, flattened)
Rémy Sigrist, PARI program for A348579
Index entries for sequences that are permutations of the natural numbers

Cf. A307720, A307730, A348246, A348409.

PARI
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See Links section.
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T(n, 1) = A348246(n).

T(n, n) = A348409(n).

A348395 Terms in A307730 in order of appearance.

Original entry on oeis.org

1, 2, 3, 6, 4, 9, 12, 8, 15, 5, 7, 14, 10, 21, 18, 24, 16, 27, 36, 20, 28, 32, 45, 25, 30, 35, 42, 49, 56, 40, 48, 63, 54, 72, 64, 81, 90, 50, 60, 70, 80, 99, 11, 13, 26, 22, 39, 33, 44, 55, 65, 52, 78, 66, 91, 77, 88, 110, 100, 121, 132, 84, 96, 108, 120, 143

Offset: 1

Rémy Sigrist, Oct 17 2021

nonn

Equivalently, terms in A348243 in order of appearance.

			Sequence A307730 starts: 1, 2, 2, 3, 3, 3, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9.
So the present sequence starts: 1, 2, 3, 6, 4, 9.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A348395

Cf. A307720, A307730, A348243.

C
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See Links section.
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cp's OEIS Frontend

A348246 Index of first occurrence of n in A307730.

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C

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

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C

Formula

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

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C

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

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A348449 Indices of records in A307730.

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C

A348247 Where prime(n) first appears in A307730.

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Python

Extensions

A348251 Numbers that take more than the minimum number of steps to appear in A307730.

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A348448 Records in A307730.

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Author

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Comments

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C

A348579 Triangle T(n, k), n > 0, 0 < k <= n, read by rows; the n-th row contains, in ascending order, the numbers m such that A307730(m) = n.

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PARI