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 10 results.

A349821 a(n) = A230624(n)/2.

Original entry on oeis.org

0, 1, 5, 7, 11, 19, 31, 47, 79, 103, 159, 191, 239, 303, 383, 479, 511, 767, 831, 863, 895, 959, 991, 1119, 1311, 1343, 1503, 1631, 1791, 1983, 2367, 2559, 2687, 2879, 2943, 3391, 4415, 4671, 4735, 4799, 5439, 6463, 6591, 6719, 7359, 9343, 11263, 12415, 21503, 23039, 36607, 50687, 51967, 56319, 56831, 63231, 64767
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2021

Keywords

Comments

The definition of A230624 implies that all its terms are even.

Crossrefs

Cf. A230624, A349820.

A349820 Primes p such that 2*p is a member of A230624.

Original entry on oeis.org

5, 7, 11, 19, 31, 47, 79, 103, 191, 239, 383, 479, 863, 991, 2687, 2879, 3391, 4799, 6719, 9343, 21503, 23039, 36607, 69119, 72959, 126719, 152063, 382463, 602111, 927743, 972799, 1096703, 1102847, 1640447, 1655807, 1966079, 3565567, 3590143, 4124671, 5402623
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2021

Keywords

Comments

It is not known if A230624 is infinite. Many of its initial terms are twice primes, so it would interesting if these primes could be characterized in some other way.
a(n)+1 typically has a slowly growing power of 2 as factor. as can be seen here: (PARI) for(k=1, #a, print1(valuation(a[k]+1,2),", ")): 1, 3, 2, 2, 5, 4, 4, 3, 6, 4, 7, 5, 5, 5, 7, 6, 6, 6, 6, 7, 10, 9, 8, 9, 8, 8, 9, 9, 12, 11, 11, 10, 10, 11, 10, 17, 11, 11, 12, 12,.. - Hugo Pfoertner Jan 03 2022

Crossrefs

Cf. A230624, A349821.

Extensions

a(29)-a(40) from Martin Ehrenstein, Jan 03 2022

A349823 First differences of A230624.

Original entry on oeis.org

2, 8, 4, 8, 16, 24, 32, 64, 48, 112, 64, 96, 128, 160, 192, 64, 512, 128, 64, 64, 128, 64, 256, 384, 64, 320, 256, 320, 384, 768, 384, 256, 384, 128, 896, 2048, 512, 128, 128, 1280, 2048, 256, 256, 1280, 3968, 3840, 2304, 18176, 3072, 27136, 28160, 2560, 8704, 1024, 12800, 3072, 6144, 2560, 7680, 4608
Offset: 1

Views

Author

N. J. A. Sloane, Dec 31 2021

Keywords

Comments

This sequence could certainly be divided by 2, just as we divided A230624 itself by 2 to get A349821. But there is a reason for not dividing this by 2: it appears that, for any power of 2, from a certain point on, the sequence is divisible by that power of 2. At present this only a conjecture. But it may provide a clue to the structure of this sequence and therefore of A230624.
For example, after 14 terms, the present sequence (as far as it is presently known) can be divided by 64, giving 3, 1, 8, 2, 1, 1, 2, 1, 4, 6, 1, 5, 4, 5, 6, 12, 6, 4, 6, 2, 14, 32, 8, 2, 2, 20, 32, 4, 4, 20, 62, 60, 36, 284, 48, 424, 440, 40, 136, 16, 200, 48, 96, 40, ..., which in turn can be divided by 2 after a further 14 terms.
So there is at least some structure here.

Links

Crossrefs

Cf. A230624, A349821.

A350607 a(n) = (A230624(n)-2)/4.

Original entry on oeis.org

0, 2, 3, 5, 9, 15, 23, 39, 51, 79, 95, 119, 151, 191, 239, 255, 383, 415, 431, 447, 479, 495, 559, 655, 671, 751, 815, 895, 991, 1183, 1279, 1343, 1439, 1471, 1695, 2207, 2335, 2367, 2399, 2719, 3231, 3295, 3359, 3679, 4671, 5631, 6207, 10751, 11519, 18303, 25343, 25983, 28159, 28415, 31615, 32383, 33919, 34559, 36479, 37631, 44927, 61055, 63359, 64127
Offset: 1

Views

Author

N. J. A. Sloane, Jan 13 2022

Keywords

Links

Crossrefs

Cf. A230624, A349821.

A349822 Irregular triangle T(n,b) (n >= 3, 2 <= b <= A230624(n)/2+1) read by rows. Let m = A230624(n). Then T(n,b) is the smallest nonnegative number k such that k+S_b(k)=m, where S_b(k) is the sum of the digits of k in base b.

Original entry on oeis.org

7, 7, 8, 7, 5, 11, 11, 10, 9, 12, 10, 7, 19, 17, 20, 17, 16, 17, 18, 15, 20, 16, 11, 35, 34, 31, 33, 29, 31, 33, 31, 28, 29, 30, 25, 32, 26, 34, 27, 36, 28, 19, 58, 58, 55, 57, 56, 55, 52, 51, 49, 51, 53, 49, 57, 52, 46, 47, 48, 49, 50, 41, 52, 42, 54, 43, 56, 44, 58, 45, 60, 46, 31
Offset: 3

Views

Author

N. J. A. Sloane, Dec 30 2021

Keywords

Comments

T(n,b) must be nonzero for all b >= 2, and so this triangle is actually the upper left corner of an array with infinitely long rows (it is believed that there are also infinitely many rows).
Since T(n,b) = m/2 for all b > m/2, we may truncate row n after m/2 terms. The rows do not change beyond that point.

Examples

			Triangle begins as follows:
   n    m   Row n
   3   10   [7, 7, 8, 7, 5],
   4   14   [11, 11, 10, 9, 12, 10, 7],
   5   22   [19, 17, 20, 17, 16, 17, 18, 15, 20, 16, 11],
   6   38   [35, 34, 31, 33, 29, 31, 33, 31, 28, 29, 30, 25, 32, 26, 34, 27, 36, 28, 19],
   7   62   [58, 58, 55, 57, 56, 55, 52, 51, 49, 51, 53, 49, 57, 52, 46, 47, 48, 49, 50, 41, 52, 42, 54, 43, 56, 44, 58, 45, 60, 46, 31],
   8   94   [90, 89, 89, 87, 87, 83, 89, 79, 83, 82, 80, 77, 86, 82, 77, 79, 81, 74, 85, 77, 89, 80, 70, 71, 72, 73, 74, 75, 76, 77, 78, 63, 80, 64, 82, 65, 84, 66, 86, 67, 88, 68, 90, 69, 92, 70, 47],
...
For n = 3, m = A230624(3) = 10, and row 3 of the triangle is [7, 7, 8, 7, 5], corresponding to the identities (where x_b is the base-b expansion of x):
   10 = 111_2 + 3 = 7 + 3,
      = 21_3 + 3 = 7 + 3
      = 20_4 + 2 = 8 + 2
      = 12_5 + 3 = 7 + 3
      = 5_b + 5  = 5 + 5 for all b >= 6.

Crossrefs

Cf. A230624.

A350601 Array read by antidiagonals: row n lists even numbers that are "generated" (in Kaprekar's sense) in all bases 2, 4, 6, ..., 2n.

Original entry on oeis.org

0, 0, 2, 0, 2, 8, 0, 2, 10, 10, 0, 2, 10, 12, 12, 0, 2, 10, 14, 14, 14, 0, 2, 10, 14, 16, 16, 16, 0, 2, 10, 14, 22, 22, 22, 20, 0, 2, 10, 14, 22, 24, 24, 24, 22, 0, 2, 10, 14, 22, 24, 28, 28, 26, 24
Offset: 1

Views

Author

N. J. A. Sloane, Jan 08 2022

Keywords

Comments

Max Alekseyev's PARI "Gen" program (see A010061) is essential for computing the rows. Cf. A349833.

Examples

			The initial rows of the array are:
  0, 2,  8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 34, 36, 38, 40, 42, 44, 50, 52,  ... [the even terms of A228082]
  0, 2, 10, 12, 14, 16, 22, 24, 26, 28, 34, 36, 38, 40, 44, 50, 58, 60, 62, 66  ... [A349831]
  0, 2, 10, 14, 16, 22, 24, 28, 34, 36, 38, 44, 50, 58, 60, 62, 66, 68, 72, 74,  ... [A349832]
  0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84,  ... [A349833]
  0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84,  ...
  0, 2, 10, 14, 22, 28, 36, ...
  0, 2, 10, 14, 22, 36, ...
  0, 2, 10, 14, 22, 36,...
  0, 2, 10, 14, 22, ...
...
The rows converge to A230624, which is
  0, 2, 10, 14, 22, 38, 62, 94, 158, 206, 318, 382, 478, 606, 766, 958, 1022, ...
The initial antidiagonals are:
  0,
  0, 2,
  0, 2, 8,
  0, 2, 10, 10,
  0, 2, 10, 12, 12,
  0, 2, 10, 14, 14, 14,
  0, 2, 10, 14, 16, 16, 16,
  0, 2, 10, 14, 22, 22, 22, 20,
  0, 2, 10, 14, 22, 24, 24, 24, 22,
  0, 2, 10, 14, 22, 24, 28, 28, 26, 24,
  ...

Links

Crossrefs

The first few rows of the array are A228082 (even terms only), A349831, A349832, and A349833.
Cf. A003052, A010061, A230624.

Extensions

[Needs checking and extending]

A349830 Intersection of A228082 and A349829.

Original entry on oeis.org

0, 2, 5, 7, 9, 10, 11, 12, 14, 16, 17, 19, 22, 24, 26, 27, 28, 29, 31, 33, 34, 36, 38, 40, 41, 43, 44, 45, 49, 50, 53, 55, 57, 58, 60, 61, 62, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 79, 81, 82, 84, 87, 89, 91, 92, 93, 94, 96, 98, 99, 101, 103, 105, 106, 108, 109, 110, 114, 115, 118, 120, 122, 123
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2022

Keywords

Comments

Numbers that are "generated" (in Kaprekar's sense) in both bases 2 and 4.

Crossrefs

Cf. A228082, A349829.
A230624 is a subsequence.

A349831 Even numbers in the intersection of A228082 and A349829.

Original entry on oeis.org

0, 2, 10, 12, 14, 16, 22, 24, 26, 28, 34, 36, 38, 40, 44, 50, 58, 60, 62, 66, 68, 70, 72, 74, 76, 82, 84, 92, 94, 96, 98, 106, 108, 110, 114, 118, 120, 122, 126, 132, 134, 136, 140, 146, 154, 156, 158, 162, 164, 170, 174, 176, 178, 186, 188, 190, 196, 198, 202, 204, 206, 210, 214, 216, 218, 222
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2022

Keywords

Comments

Even numbers that are "generated" (in Kaprekar's sense) in both bases 2 and 4.

Crossrefs

Cf. A003052, A228082, A349829, A349830, A349832.
A230624 is a subsequence.
A row of A350601.

A349832 Even numbers that are "generated" (in Kaprekar's sense) in all three bases 2, 4, and 6.

Original entry on oeis.org

0, 2, 10, 14, 16, 22, 24, 28, 34, 36, 38, 44, 50, 58, 60, 62, 66, 68, 72, 74, 76, 82, 84, 92, 94, 96, 98, 106, 108, 110, 118, 120, 122, 126, 132, 134, 136, 140, 146, 154, 156, 158, 162, 164, 170, 176, 178, 186, 196, 198, 202, 206, 210, 214, 216, 222, 228, 234, 238, 244, 246, 252, 256, 258, 260
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2022

Keywords

Comments

Using Max Alekseyev's PARI "Gen" program (see A010061), we run
vector(500,k,length(Gen(k,2))),
vector(500,k,length(Gen(k,4))), and
vector(500,k,length(Gen(k,6)))
to find the numbers that are generated in bases 2, 4, and 6, and then take the even numbers that are common to all three lists.

Crossrefs

Cf. A003052, A010061, A228082, A349829, A349830, A349831, A349833.
A230624 is a subsequence.
A row of A350601.

A349833 Even numbers that are "generated" (in Kaprekar's sense) in all four bases 2, 4, 6, and 8.

Original entry on oeis.org

0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84, 92, 94, 96, 98, 106, 110, 118, 120, 122, 132, 134, 136, 140, 154, 156, 158, 162, 170, 176, 178, 186, 196, 198, 206, 210, 214, 216, 222, 228, 234, 244, 246, 252, 258, 260, 262, 264, 268, 274, 284, 286
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2022

Keywords

Comments

Using Max Alekseyev's PARI "Gen" program (see A010061), we run
vector(500,k,length(Gen(k,2))),
vector(500,k,length(Gen(k,4))),
vector(500,k,length(Gen(k,6))),
vector(500,k,length(Gen(k,8))),
to find the numbers that are generated in bases 2, 4, 6, and 8, and then take the even numbers that are common to all four lists.

Links

Crossrefs

Cf. A003052, A010061, A228082, A349829, A349830, A349831, A349832.
A230624 is a subsequence.
A row of A350601.
Showing 1-10 of 10 results.

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

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