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.

Previous Showing 11-17 of 17 results.

A284579 Carryless base-2 product (A048720) of run lengths in binary representation of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 1, 2, 4, 2, 3, 4, 4, 3, 2, 4, 2, 1, 2, 3, 6, 4, 2, 4, 6, 3, 4, 5, 5, 4, 3, 6, 4, 2, 4, 6, 3, 2, 1, 2, 4, 2, 3, 4, 8, 6, 4, 8, 4, 2, 4, 6, 5, 6, 3, 6, 8, 4, 5, 6, 6, 5, 4, 8, 6, 3, 6, 5, 6, 4, 2, 4, 8, 4, 6, 8, 4, 3, 2, 4, 2, 1, 2, 3, 6, 4, 2, 4, 6, 3, 4, 5, 10, 8, 6, 12, 8, 4, 8, 12, 6, 4, 2, 4, 8, 4, 6, 8, 12, 5, 6, 12, 6, 3, 6
Offset: 0

Views

Author

Antti Karttunen, Apr 14 2017

Keywords

Examples

			For n=56, A007088(56) = "111000" in binary, we do carryless multiplication (in base-2) of 3 and 3, thus a(56) = A048720(3,3) = 5.

Links

Crossrefs

Cf. A000975 (positions of ones).
Cf. A007088, A048720, A284580, A284581.
Differs from A167489 for the first time at n=56, where a(56) = 5, while A167489(56) = 9.

Programs

  • Scheme
    (define (A284579 n) (reduce A048720bi 1 (binexp->runcount1list n))) ;; Where A048720bi is a two-argument function implementing carryless binary product, A048720. For binexp->runcount1list see A167489.

Formula

A284581(n) = n - a(n).

A227352 Permutation of nonnegative integers: map each number by lengths of runs in its binary representation to the number in whose once left-shifted Zeckendorf representation occurs the same run lengths (in the same order) as the lengths of consecutive blocks of zeros.

Original entry on oeis.org

0, 1, 4, 2, 7, 12, 6, 3, 11, 19, 33, 20, 10, 17, 9, 5, 18, 30, 51, 31, 54, 88, 53, 32, 16, 27, 46, 28, 15, 25, 14, 8, 29, 48, 80, 49, 83, 135, 82, 50, 87, 142, 232, 143, 86, 140, 85, 52, 26, 43, 72, 44, 75, 122, 74, 45, 24, 40, 67, 41, 23, 38, 22, 13, 47, 77
Offset: 0

Views

Author

Antti Karttunen, Jul 08 2013

Keywords

Comments

See the comments at the inverse permutation A227351 where the idea behind this mapping is explained.

Links

Crossrefs

Inverse permutation: A227351. Cf. A048680, A003188.

Programs

Formula

a(n) = A048680(A003188(n)). [The defining formula]
Moreover, this permutation effects the following correspondences:
For n>=1 A000523(n) = A102364(a(n)).
For all n, A167489(n) = A227355(a(2n+1)).

A227355 Product of run lengths in Zeckendorf representation of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 2, 1, 5, 4, 3, 4, 2, 3, 2, 1, 6, 5, 4, 6, 3, 6, 4, 2, 4, 3, 2, 2, 1, 7, 6, 5, 8, 4, 9, 6, 3, 8, 6, 4, 4, 2, 5, 4, 3, 4, 2, 3, 2, 1, 8, 7, 6, 10, 5, 12, 8, 4, 12, 9, 6, 6, 3, 10, 8, 6, 8, 4, 6, 4, 2, 6, 5, 4, 6, 3, 6, 4, 2, 4, 3
Offset: 0

Views

Author

Antti Karttunen, Jul 08 2013

Keywords

Comments

The same sequence also gives the product for the lengths of zero-runs only, as by definition, no two consecutive 1's can occur in Fibonacci number system (aka Zeckendorf representation), thus any 1's present contribute just *1 to the total product.

Links

Crossrefs

Cf. A167489, A003714, A102364, A014417, A227350.

Programs

Formula

a(n) = A167489(A003714(n)) = A227350(A003714(n)).
a(A227352(A005408(n))) = A167489(n).
For n>= 3, a(A000045(n)) = n-2.

A284582 a(n) = gcd(A227349(n), A227350(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Apr 14 2017

Keywords

Links

Crossrefs

Cf. A167489, A227349, A227350, A284583.

Programs

Formula

a(n) = gcd(A227349(n), A227350(n)).
A167489(n) = a(n) * A284583(n).

A284583 a(n) = lcm(A227349(n), A227350(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 3, 6, 2, 2, 4, 6, 3, 4, 5, 5, 4, 3, 6, 4, 2, 2, 6, 3, 2, 1, 2, 2, 2, 3, 4, 4, 6, 2, 4, 2, 2, 4, 6, 3, 6, 3, 6, 4, 4, 5, 6, 6, 5, 4, 4, 6, 3, 6, 3, 6, 4, 2, 2, 4, 2, 6, 4, 4, 3, 2, 2, 2, 1, 2, 3, 6, 2, 2, 4, 6, 3, 4, 5, 10, 4, 6, 12, 4, 2, 4, 6, 6, 2, 2, 4, 4, 4, 6, 8, 12, 3, 6, 6, 6, 3, 6
Offset: 0

Views

Author

Antti Karttunen, Apr 14 2017

Keywords

Links

Crossrefs

Cf. A167489, A227349, A227350, A284583.

Programs

Formula

a(n) = lcm(A227349(n), A227350(n)).
A167489(n) = a(n) * A284582(n).

A167490 a(n) = Smallest number with binary run length product = n.

Original entry on oeis.org

0, 3, 7, 12, 31, 24, 127, 48, 56, 96, 2047, 99, 8191, 384, 224, 195, 131071, 199, 524287, 387, 896, 6144, 8388607, 391, 992, 24576, 455, 1539, 536870911, 775
Offset: 1

Views

Author

Andrew Weimholt, Nov 05 2009

Keywords

Comments

a(p) = 2^p - 1 for prime p

Examples

			a(4) = 12, because 12 is the smallest number with a binary run length product of 4.
12 decimal = 1100 binary. Run lengths in binary are 2,2, and 2x2 = 4.

Crossrefs

Cf. A167489 - Product of run length in binary representation of n
Cf. A167491 - Numbers in this sequence sorted in ascending order

A167491 Members of A167490 sorted in ascending order.

Original entry on oeis.org

0, 3, 7, 12, 24, 31, 48, 56, 96, 99, 127, 195, 199, 224, 384, 387, 391, 455, 775, 780, 792, 896, 992, 1539, 1548, 1560, 1592, 1799, 2047, 3079, 3096, 3103, 3120, 3128, 3640, 3968, 6144, 6156, 6192, 6200, 6243, 6343, 7175, 7199, 8191, 12312, 12319, 12384
Offset: 1

Views

Author

Andrew Weimholt, Nov 05 2009

Keywords

Comments

a(n) is the smallest number with the product of its binary run lengths = A167489(a(n))

Examples

			12 is in the sequence because the product of the run lengths in the binary representation of 12 is 4, and no number less than 12 has a binary run length product of 4.

Crossrefs

Cf. A167489 - Product of run lengths in the binary representation of n
Cf. A167490 - Smallest number with binary run length product = n
Previous Showing 11-17 of 17 results.

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