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.

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A093327 Smallest sequence of numbers having in their binary representation alternately more (>) or not-more (<=) binary ones than zeros.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 11, 12, 13, 16, 19, 20, 21, 24, 25, 32, 39, 40, 43, 44, 45, 48, 51, 52, 53, 56, 57, 64, 71, 72, 75, 76, 77, 80, 83, 84, 85, 88, 89, 96, 99, 100, 101, 104, 105, 112, 113, 128, 143, 144, 151, 152, 155, 156, 157, 160, 167, 168, 171, 172, 173, 176
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 12 2004

Keywords

Comments

A000120(a(n)) < A023416(a(n)) iff A000120(a(n+1))>=A023416(a(n+1));
A037861(a(2*n)) >= 0, A037861(a(2*n+1)) < 0.

Crossrefs

A117123 n minus the number of 0's in binary expansion of n.

Original entry on oeis.org

1, 1, 3, 2, 4, 5, 7, 5, 7, 8, 10, 10, 12, 13, 15, 12, 14, 15, 17, 17, 19, 20, 22, 21, 23, 24, 26, 26, 28, 29, 31, 27, 29, 30, 32, 32, 34, 35, 37, 36, 38, 39, 41, 41, 43, 44, 46, 44, 46, 47, 49, 49, 51, 52, 54, 53, 55, 56, 58, 58, 60, 61, 63, 58, 60, 61, 63, 63, 65, 66, 68, 67, 69
Offset: 1

Views

Author

Robert G. Wilson v, Apr 19 2006

Keywords

Crossrefs

Programs

  • Mathematica
    f[n_] := n - Count[IntegerDigits[n, 2], 0]; Array[f, 73]
    Table[n-DigitCount[n,2,0],{n,80}] (* Harvey P. Dale, Dec 24 2015 *)

A301895 a(n) = (number of 1's in binary expansion of n)^(number of 0's in binary expansion of n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 4, 4, 3, 4, 3, 3, 1, 1, 8, 8, 9, 8, 9, 9, 4, 8, 9, 9, 4, 9, 4, 4, 1, 1, 16, 16, 27, 16, 27, 27, 16, 16, 27, 27, 16, 27, 16, 16, 5, 16, 27, 27, 16, 27, 16, 16, 5, 27, 16, 16, 5, 16, 5, 5, 1, 1, 32, 32, 81, 32, 81, 81, 64, 32, 81, 81, 64, 81, 64, 64, 25, 32
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 28 2018

Keywords

Comments

Union of A000079 and A000225 without zero gives positions of ones.

Examples

			+---+------+---+---+---------+
| n | bin. |1's|0's|  a(n)   |
+---+------+---+---+---------+
| 0 |    0 | 0 | 1 | 0^1 = 0 |
| 1 |    1 | 1 | 0 | 1^0 = 1 |
| 2 |   10 | 1 | 1 | 1^1 = 1 |
| 3 |   11 | 2 | 0 | 2^0 = 1 |
| 4 |  100 | 1 | 2 | 1^2 = 1 |
| 5 |  101 | 2 | 1 | 2^1 = 2 |
| 6 |  110 | 2 | 1 | 2^1 = 2 |
| 7 |  111 | 3 | 0 | 3^0 = 1 |
| 8 | 1000 | 1 | 3 | 1^3 = 1 |
| 9 | 1001 | 2 | 2 | 2^2 = 4 |
+---+------+---+---+---------+
bin. - n written in base 2;
1's - number of 1's in binary expansion of n;
0's - number of 0's in binary expansion of n.
		

Crossrefs

Programs

  • Mathematica
    DigitCount[Range[0, 80], 2, 1]^DigitCount[Range[0, 80], 2, 0]

Formula

a(n) = A000120(n)^A023416(n).
a(A000051(n)) = A011782(n).
Previous Showing 41-43 of 43 results.