A165820 -id:A165820 - cp's OEIS frontend

A165819 a(n) = the smallest positive integer that, when written in binary, contains both binary n and binary 2n-1 as substrings.

1, 6, 11, 28, 37, 22, 29, 120, 137, 83, 43, 92, 205, 59, 61, 496, 529, 291, 595, 167, 169, 86, 183, 376, 793, 410, 437, 220, 925, 123, 125, 2016, 2081, 1091, 2211, 583, 293, 1227, 2471, 2536, 2601, 339, 171, 2796, 729, 366, 751, 1520, 3121, 1586, 1637, 1652

Offset: 1

Leroy Quet, Sep 28 2009

If a(n) in binary was instead to contain binary n and binary 2n+1 as substrings, then a(n) would = 2n+1 for all n.

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

Cf. A165820, A165821, A165822, A294977.

a(n) = A294977(n, 2*n-1). - Rémy Sigrist, Mar 03 2018

More terms from Carl R. White, Oct 06 2009

A165821 a(n) = the smallest positive integer that, when written in binary, contains both binary n and the binary representation of the n-th prime as substrings.

Original entry on oeis.org

2, 6, 11, 28, 11, 13, 71, 152, 151, 58, 95, 101, 105, 174, 47, 848, 571, 978, 1075, 327, 1173, 719, 1335, 1432, 1561, 1626, 827, 860, 237, 241, 127, 131, 4233, 139, 4501, 1175, 4765, 2467, 167, 1384, 5299, 362, 703, 705, 1581, 1479, 3027, 3568, 227, 229

Offset: 1

Leroy Quet, Sep 28 2009

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

Cf. A000040, A165819, A165820, A165822, A294977.

a(n) = A294977(n, A000040(n)). - Rémy Sigrist, Mar 03 2018

More terms from Carl R. White, Oct 06 2009

A165822 a(n) = the smallest positive integer that, when written in binary, contains both binary n and the binary representation of the n-th Fibonacci number as substrings.

Original entry on oeis.org

1, 2, 6, 12, 5, 24, 29, 168, 137, 442, 89, 400, 233, 377, 7778, 15792, 1597, 18968, 66899, 6765, 10946, 181551, 28657, 1483800, 206097, 121393, 458562, 317811, 514229, 3977768, 21540319, 2178309, 56393249, 72811751, 9227465, 618910128

Offset: 1

Leroy Quet, Sep 28 2009

Rémy Sigrist, Table of n, a(n) for n = 1..1000

Cf. A000045, A165819, A165820, A165821, A294977.

a(n) = A294977(n, A000045(n)). - Rémy Sigrist, Mar 03 2018

More terms from Carl R. White, Oct 06 2009

A294977 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is the smallest positive integer that, when written in binary, contains both binary n and binary k as substrings.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 4, 3, 4, 5, 6, 5, 12, 12, 5, 6, 7, 6, 11, 4, 11, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 7, 12, 5, 12, 7, 8, 9, 10, 9, 24, 28, 13, 13, 28, 24, 9, 10, 11, 10, 19, 8, 23, 6, 23, 8, 19, 10, 11, 12, 11, 26, 9, 40, 14, 14, 40, 9, 26

Offset: 1

Rémy Sigrist, Mar 02 2018

When computing T(n, k), we have three situations:
- the binary representation of n appears in the binary representation of k or vice versa; then T(n, k) = max(n, k); for example T(1, 2) = 2,
- otherwise a strict suffix of the binary representation of n equals a strict prefix of the binary representation of k or vice versa; then max(n, k) < T(n, k) < min(A163621(n, k), A163621(k, n)); for example T(2, 3) = 6,
- otherwise the binary representations of n and of k do not overlap; then T(n, k) = min(A163621(n, k), A163621(k, n)); for example T(10, 12) = 172.

			Array T(n, k) begins (in decimal):
  n\k|    1    2    3    4    5    6    7    8    9   10   11   12
  ---+------------------------------------------------------------
    1|    1    2    3    4    5    6    7    8    9   10   11   12
    2|    2    2    6    4    5    6   14    8    9   10   11   12
    3|    3    6    3   12   11    6    7   24   19   26   11   12
    4|    4    4   12    4   20   12   28    8    9   20   44   12
    5|    5    5   11   20    5   13   23   40   37   10   11   44
    6|    6    6    6   12   13    6   14   24   25   26   22   12
    7|    7   14    7   28   23   14    7   56   39   58   23   28
    8|    8    8   24    8   40   24   56    8   72   40   88   24
Array T(n, k) begins (in binary):
   n\k|     1    10     11    100     101    110      111    1000     1001    1010
  ----+---------------------------------------------------------------------------
     1|     1    10     11    100     101    110      111    1000     1001    1010
    10|    10    10    110    100     101    110     1110    1000     1001    1010
    11|    11   110     11   1100    1011    110      111   11000    10011   11010
   100|   100   100   1100    100   10100   1100    11100    1000     1001   10100
   101|   101   101   1011  10100     101   1101    10111  101000   100101    1010
   110|   110   110    110   1100    1101    110     1110   11000    11001   11010
   111|   111  1110    111  11100   10111   1110      111  111000   100111  111010
  1000|  1000  1000  11000   1000  101000  11000   111000    1000  1001000  101000

Rémy Sigrist, PARI program for A294977

Cf. A000040, A000045, A163621, A165819, A165820, A165821, A165822.

PARI
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See Links section.
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T(n, n) = n.
T(n, 1) = n.
T(n, k) = T(k, n).
T(T(n, k), k) = T(n, k) (for any fixed n > 0, the function k -> T(n, k) is a projection).
A165819(n) = T(n, 2*n-1).
A165820(n) = T(n, n^2).
A165821(n) = T(n, A000040(n)).
A165822(n) = T(n, A000045(n)).
T(n, k) >= n with equality iff the binary representation of k appears in the binary representation of n.
T(n, k) >= max(n, k).
T(n, k) <= min(A163621(n, k), A163621(k, n)) with equality iff the binary representations of n and of k do not overlap.

cp's OEIS Frontend

A165819 a(n) = the smallest positive integer that, when written in binary, contains both binary n and binary 2n-1 as substrings.

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A165821 a(n) = the smallest positive integer that, when written in binary, contains both binary n and the binary representation of the n-th prime as substrings.

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A165822 a(n) = the smallest positive integer that, when written in binary, contains both binary n and the binary representation of the n-th Fibonacci number as substrings.

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A294977 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is the smallest positive integer that, when written in binary, contains both binary n and binary k as substrings.

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