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 1-4 of 4 results.

A284581 a(n) = n minus (carryless base-2 product of run lengths in binary representation of n) = n - A284579(n).

Original entry on oeis.org

0, 1, 1, 2, 4, 4, 4, 5, 7, 9, 9, 8, 11, 11, 11, 12, 14, 16, 15, 18, 20, 20, 20, 18, 21, 24, 23, 22, 26, 26, 26, 27, 29, 31, 29, 32, 35, 34, 33, 37, 39, 41, 41, 40, 43, 43, 43, 40, 43, 46, 43, 48, 51, 50, 49, 51, 51, 55, 53, 52, 57, 57, 57, 58, 60, 62, 59, 62, 66, 64, 66, 66, 69, 72, 71, 68, 73, 72, 71, 76, 78, 80, 79, 82, 84, 84, 84, 82, 85, 88, 87, 86
Offset: 1

Views

Author

Antti Karttunen, Apr 14 2017

Keywords

Links

Crossrefs

Cf. A048720, A284579.
Cf. A227190, A284560, A283972 for similar sequences.
Differs from A227190 for the first time at n=56, where a(56)=51, while A227190(56)=47.

Programs

Formula

a(n) = n - A284579(n).

A167489 Product 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, 9, 6, 3, 6, 8, 4, 5, 6, 6, 5, 4, 8, 6, 3, 6, 9, 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
Offset: 0

Views

Author

Andrew Weimholt, Nov 05 2009

Keywords

Examples

			a(56) = 9, because 56 in binary is written 111000 giving the run lengths 3,3 and 3x3 = 9.
a(99) = 12, because 99 in binary is written 1100011 giving the run lengths 2,3,2, and 2x3x2 = 12.

Links

Crossrefs

Row products of A101211 and A227736 (for n > 0).
Cf. A167490 (smallest number with binary run length product = n).
Cf. A167491 (members of A167490 sorted in ascending order).
Cf. A227355, A227184, A227190.
Cf. A227349, A227350, A227352, A246588, A284558, A284559, A284582, A284583.
Differs from similar A284579 for the first time at n=56, where a(56) = 9, while A284579(56) = 5.

Programs

  • Haskell
    import Data.List (group)
a167489 = product . map length . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013
  • Mathematica
    Table[ Times @@ (Length /@ Split[IntegerDigits[n, 2]]), {n, 0, 100}](* Olivier Gérard, Jul 05 2013 *)
  • PARI
    a(n) = {my(p=1, b=n%2, i=0); while(n!=0, n=n>>1; i=i+1; if((n%2)!=b, p=p*i; i=0; b=n%2)); p} \\ Indranil Ghosh, Apr 17 2017, after the Python Program by Antti Karttunen
  • Python
    def A167489(n):
  '''Product of run lengths in binary representation of n.'''
  p = 1
  b = n%2
  i = 0
  while (n != 0):
    n >>= 1
    i += 1
    if ((n%2) != b):
      p *= i
      i = 0
      b = n%2
  return(p)
# Antti Karttunen, Jul 24 2013 (Cf. Python program for A227184).
  • Scheme
    (define (A167489 n) (apply * (binexp->runcount1list n)))
(define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2)))))))
;; Antti Karttunen, Jul 05 2013

Formula

a(n) = A227349(n) * A227350(n) = A227355(A227352(2n+1)). - Antti Karttunen, Jul 25 2013
a(n) = A284558(n) * A284559(n) = A284582(n) * A284583(n). - Antti Karttunen, Apr 16 2017

A284559 a(n) = LCM 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, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 3, 6, 2, 2, 2, 6, 3, 4, 5, 5, 4, 3, 6, 2, 2, 2, 6, 3, 2, 1, 2, 2, 2, 3, 4, 4, 6, 2, 2, 2, 2, 2, 6, 3, 6, 3, 6, 4, 4, 5, 6, 6, 5, 4, 4, 6, 3, 6, 3, 6, 2, 2, 2, 2, 2, 6, 4, 4, 3, 2, 2, 2, 1, 2, 3, 6, 2, 2, 2, 6, 3, 4, 5, 10, 4, 6, 6, 2, 2, 2, 6, 6, 2, 2, 2, 2, 2, 6, 4, 12, 3, 6, 6, 6, 3, 6, 3
Offset: 0

Views

Author

Antti Karttunen, Apr 14 2017

Keywords

Examples

			For n=12, A007088(12) = "1100" in binary, the run lengths are [2,2], thus a(12) = lcm(2,2) = 2.

Links

Crossrefs

Cf. A000975 (positions of ones).
Cf. A007088, A167489, A227349, A284558, A284569, A284579.

Programs

  • Python
    from math import lcm
from itertools import groupby
def a(n): return lcm(*(len(list(g)) for k, g in groupby(bin(n)[2:])))
print([a(n) for n in range(87)]) # Michael S. Branicky, Oct 15 2022
  • Scheme
    (define (A284559 n) (apply lcm (binexp->runcount1list n)))
;; Or:
(define (A284559 n) (reduce lcm 1 (binexp->runcount1list n))) ;; For binexp->runcount1list, see the Program section of A227349.

Formula

a(n) = A167489(n) / A284558(n).

A284580 Carryless base-2 product (A048720) of lengths of runs of 1-bits in binary representation of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 14 2017

Keywords

Examples

			a(119) = 5, as 119 is "1110111" in binary, and A048720(3,3) = 5.

Links

Crossrefs

Cf. A003714 (positions of ones).
Cf. A048720, A284579.
Differs from similar A227349 for the first time at n=119, where a(119) = 5, while A227349(119) = 9.

Programs

  • Scheme
    (define (A284580 n) (reduce A048720bi 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2))))) ;; Where A048720bi is a two-argument function implementing carryless binary product, A048720. For bisect and binexp->runcount1list, see under A227349.
Showing 1-4 of 4 results.

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