cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

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

Views

Author

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

Keywords

Comments

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.

Links

Crossrefs

Cf. A307720, A307730, A348245, A348246, A348409.
Records: A348255, A348256.

A348254 Records in A348409.

Original entry on oeis.org

1, 3, 6, 11, 51, 58, 69, 1384, 1397, 3754, 3773, 6702, 6731, 84656, 84693, 89520, 89563, 91878, 91931, 173150, 173211, 192948, 193019, 524659, 524738, 865383, 865472, 876639, 876740, 890581, 890688, 904381, 904494, 918985, 919116, 1629115, 1629254, 1647255, 1647406, 1666931, 1667094, 1687929, 1688102
Offset: 1

Views

Author

N. J. A. Sloane, Oct 21 2021

Keywords

Crossrefs

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

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

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Apr 25 2019

Keywords

Comments

For all positive integers n, n appears n times.

Examples

			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

Links

Crossrefs

Cf. A002024, A088178, A307720.
Cf. A348579 (indices of occurrence of each number), A348246 (first occurrence of each number), A348409 (last occurrence).

Programs

  • PARI
    \\ See Links section.
  • Python
    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

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A307720, A307730, A307630, A348409.
Records: A348252, A348253.

Programs

  • C
    See Links section.

Formula

A348409(n) - a(n) >= n - 1.

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

Views

Author

N. J. A. Sloane, Oct 21 2021

Keywords

Links

Crossrefs

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

Programs

  • C
    // See Links section.

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

Views

Author

N. J. A. Sloane, Oct 21 2021

Keywords

Links

Crossrefs

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

Programs

  • C
    See Links section.

Extensions

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

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

Views

Author

N. J. A. Sloane, Oct 21 2021

Keywords

Crossrefs

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

A348250 a(n) = A348249(n) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 8, 0, 10, 37, 0, 0, 0, 0, 0, 32, 0, 0, 16, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 80, 0, 34, 133, 0, 0, 0, 0, 0, 326, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 88, 2605, 46, 0, 0, 0, 0, 98, 0, 90, 0, 0, 0, 0, 0, 0, 64, 0, 62, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 228, 82, 3139, 0, 0, 260, 1135
Offset: 1

Views

Author

N. J. A. Sloane, Oct 21 2021

Keywords

Comments

A348249(n) is at least n (that is at the heart of the definition of A307720). Here we examine the difference.

Links

Crossrefs

Cf. A307720, A307730, A348246, A348249, A348409, A348251.

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

Views

Author

N. J. A. Sloane, Oct 21 2021

Keywords

Comments

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.

Links

Crossrefs

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

A348255 Records in A348249.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 11, 20, 24, 52, 53, 116, 172, 371, 2662, 3226, 4800, 9546, 9804, 14171, 69148, 71863, 325494, 333873, 698509, 2262261, 15387377, 40004428, 40813283, 64314104, 550623089, 847235953, 897871194, 1259250335
Offset: 1

Views

Author

N. J. A. Sloane, Oct 21 2021

Keywords

Crossrefs

Cf. A307720, A307730, A348246, A348409, A348249, A348252-A348256.
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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