A331857 -id:A331857 - cp's OEIS frontend

A331855 a(n) is the number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then reversing those blocks.

1, 1, 2, 1, 3, 3, 3, 1, 4, 6, 5, 4, 5, 4, 4, 1, 5, 10, 9, 9, 8, 8, 9, 5, 7, 9, 8, 5, 7, 5, 5, 1, 6, 15, 14, 16, 12, 16, 18, 12, 11, 16, 13, 12, 15, 13, 14, 6, 9, 16, 15, 13, 13, 12, 12, 6, 10, 12, 11, 6, 9, 6, 6, 1, 7, 21, 20, 25, 18, 27, 30, 22, 16, 27, 25

Offset: 0

Rémy Sigrist, Jan 29 2020

			For n = 6:
- the binary representation of 6 is "110",
- we can split it in 4 ways:
      "110" -> "011" -> 3
      "1" and "10" -> "1" and "01" -> 5
      "11" and "0" -> "11" and "0" -> 6
      "1" and "1" and "0" -> "1" and "1" and "0" -> 6
- we have 3 distinct values,
- hence a(6) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Rémy Sigrist, PARI program for A331855
Index entries for sequences related to binary expansion of n

See A331851 for similar sequences.
See A331856 and A331857 for the least and greatest values, respectively.
Cf. A000096, A000217.

PARI
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See Links section.
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a(2^k-1) = 1 for any k >= 0.
a(2^k) = k+1 for any k >= 0.
a(2^k+1) = A000217(k) for any k > 0.
a(2^k+2) = A000096(k-1) for any k > 3.
a(2^k+3) = (k-1)^2 for any k > 1.

A331856 a(n) is the least value obtained by partitioning the binary representation of n into consecutive blocks, and then reversing those blocks.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 5, 7, 3, 7, 7, 15, 1, 3, 5, 7, 5, 11, 11, 15, 3, 7, 11, 15, 7, 15, 15, 31, 1, 3, 5, 7, 9, 11, 11, 15, 5, 11, 21, 23, 13, 23, 23, 31, 3, 7, 13, 15, 11, 23, 27, 31, 7, 15, 23, 31, 15, 31, 31, 63, 1, 3, 5, 7, 9, 11, 11, 15, 9, 19, 21

Offset: 0

Rémy Sigrist, Jan 29 2020

			For n = 6:
- the binary representation of 6 is "110",
- we can split it in 4 ways:
      "110" -> "011" -> 3
      "1" and "10" -> "1" and "01" -> 5
      "11" and "0" -> "11" and "0" -> 6
      "1" and "1" and "0" -> "1" and "1" and "0" -> 6
- we have 3 distinct values, the least being 3,
- hence a(6) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Rémy Sigrist, PARI program for A331856
Index entries for sequences related to binary expansion of n

See A331855 for the number of distinct values, and A331857 for the greatest value.

Cf. A000225, A023416, A038573.

PARI
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See Links section.
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a(n)^A023416(n) = A038573(n) (where a^k denotes the k-th iterate of a).

a(n) <= n with equality iff n belongs to A000225.

cp's OEIS Frontend

A331855 a(n) is the number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then reversing those blocks.

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