A227350 -id:A227350 - cp's OEIS frontend

A227349 Product of lengths of runs of 1-bits in binary representation of n.

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, 9, 4

Offset: 0

Antti Karttunen, Jul 08 2013

This is the Run Length Transform of S(n) = {0, 1, 2, 3, 4, 5, 6, ...}. The Run Length Transform of a sequence {S(n), n >= 0} is defined to be the sequence {T(n), n >= 0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0) = 1 (the empty product). - N. J. A. Sloane, Sep 05 2014

Like all run length transforms also this sequence satisfies for all i, j: A278222(i) = A278222(j) => a(i) = a(j). - Antti Karttunen, Apr 14 2017

			a(0) = 1, as zero has no runs of 1's, and an empty product is 1.
a(1) = 1, as 1 is "1" in binary, and the length of that only 1-run is 1.
a(2) = 1, as 2 is "10" in binary, and again there is only one run of 1-bits, of length 1.
a(3) = 2, as 3 is "11" in binary, and there is one run of two 1-bits.
a(55) = 6, as 55 is "110111" in binary, and 2 * 3 = 6.
a(119) = 9, as 119 is "1110111" in binary, and 3 * 3 = 9.
From _Omar E. Pol_, Feb 10 2015: (Start)
Written as an irregular triangle in which row lengths is A011782:
  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;
  ...
Right border gives A028310: 1 together with the positive integers. (End)
From _Omar E. Pol_, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s, r, k) as shown below:
  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;
  ...
Apart from the initial 1, we have that T(s, r, k) = T(s+1, r, k). (End)

Antti Karttunen, Table of n, a(n) for n = 0..8192
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences computed with run length transform

Cf. A003714 (positions of ones), A005361, A005940.
Cf. A000120 (sum of lengths of runs of 1-bits), A167489, A227350, A227193, A278222, A245562, A284562, A284569, A283972, A284582, A284583.
Run Length Transforms of other sequences: A246588, A246595, A246596, A246660, A246661, A246674.
Differs from similar A284580 for the first time at n=119, where a(119) = 9, while A284580(119) = 5.

a:= proc(n) local i, m, r; m, r:= n, 1;
      while m>0 do
        while irem(m, 2, 'h')=0 do m:=h od;
        for i from 0 while irem(m, 2, 'h')=1 do m:=h od;
        r:= r*i
      od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013
ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n, base, 2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
   elif out1 = 0 and t1[i] = 1 then c:=c+1;
   elif out1 = 1 and t1[i] = 0 then c:=c;
   elif out1 = 0 and t1[i] = 0 then lis:=[c, op(lis)]; out1:=1; c:=0;
   fi;
   if i = L1 and c>0 then lis:=[c, op(lis)]; fi;
                   od:
a:=mul(i, i in lis);
ans:=[op(ans), a];
od:
ans;  # N. J. A. Sloane, Sep 05 2014

onBitRunLenProd[n_] := Times @@ Length /@ Select[Split @ IntegerDigits[n, 2], #[[1]] == 1 & ]; Array[onBitRunLenProd, 100, 0] (* Jean-François Alcover, Mar 02 2016 *)

from operator import mul
from functools import reduce
from re import split
def A227349(n):
    return reduce(mul, (len(d) for d in split('0+',bin(n)[2:]) if d)) if n > 0 else 1 # Chai Wah Wu, Sep 07 2014

# uses[RLT from A246660]
A227349_list = lambda len: RLT(lambda n: n, len)
A227349_list(88) # Peter Luschny, Sep 07 2014

(define (A227349 n) (apply * (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
(define (bisect lista parity) (let loop ((lista lista) (i 0) (z (list))) (cond ((null? lista) (reverse! z)) ((eq? i parity) (loop (cdr lista) (modulo (1+ i) 2) (cons (car lista) z))) (else (loop (cdr lista) (modulo (1+ i) 2) z)))))
(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)))))))

A167489(n) = a(n) * A227350(n).
A227193(n) = a(n) - A227350(n).
a(n) = Product_{i in row n of table A245562} i. - N. J. A. Sloane, Aug 10 2014
From Antti Karttunen, Apr 14 2017: (Start)
a(n) = A005361(A005940(1+n)).
a(n) = A284562(n) * A284569(n).
A283972(n) = n - a(n).
(End)
a(4n+1) = a(2n) = a(n). If n is odd, then a(4n+3) = 2*a(2n+1)-a(n). If n is even, then a(4n+3) = 2*a(2n+1) = 2*a(n/2). - Chai Wah Wu, Jul 17 2025

Data section extended up to term a(120) by Antti Karttunen, Apr 14 2017

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

Andrew Weimholt, Nov 05 2009

			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.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

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.

import Data.List (group)
a167489 = product . map length . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013

Table[ Times @@ (Length /@ Split[IntegerDigits[n, 2]]), {n, 0, 100}](* Olivier Gérard, Jul 05 2013 *)

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

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).

(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

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

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

Antti Karttunen, Jul 08 2013

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.

Antti Karttunen, Table of n, a(n) for n = 0..10946

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

(define (A227355 n) (A167489 (A003714 n)))(define (A227355v2 n) (A227350 (A003714 n))) ;; Alternative definition.

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

A227193 Difference of (product of runlengths of 1-bits) and (product of runlengths of 0-bits) in binary representation of n.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 1, 2, -2, -1, 0, 1, 0, 1, 2, 3, -3, -2, -1, 0, -1, 0, 1, 2, -1, 0, 1, 3, 1, 2, 3, 4, -4, -3, -2, -1, -3, -1, 0, 1, -2, -1, 0, 1, 0, 1, 2, 3, -2, -1, 0, 2, 0, 1, 3, 5, 0, 1, 2, 5, 2, 3, 4, 5, -5, -4, -3, -2, -5, -2, -1, 0, -5, -3, -1, 0, -2, 0

Offset: 0

Antti Karttunen, Jul 08 2013

sign

The sequence seems to consist of palindromic subsequences centered around each (2^k)-1 and 2^k (with end points near the terms of A000975), which is easily explained by symmetric pairing of binary expansion of n and its complement.

Antti Karttunen, Table of n, a(n) for n = 0..10922

Cf. A145037, A227190.

a:= proc(n) local i, j, m, r, s; m, r, s:= n, 1, 1;
      while m>0 do
        for i from 0 while irem(m, 2, 'h')=0 do m:=h od;
        for j from 0 while irem(m, 2, 'h')=1 do m:=h od;
        r, s:= r*j, s*max(i, 1)
      od; r-s
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013

a[n_] := With[{s = Split @ IntegerDigits[n, 2]}, Times @@ Length /@ Select[ s, First[#]==1&] - Times @@ Length /@ Select[s , First[#]==0&]]; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 28 2016 *)

(define (A227193 n) (- (A227349 n) (A227350 n)))

a(n) = A227349(n) - A227350(n).

cp's OEIS Frontend

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

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

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A227349 Product of lengths of runs of 1-bits in binary representation of n.

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A167489 Product of run lengths in binary representation of n.

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A227355 Product of run lengths in Zeckendorf representation of n.

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A227193 Difference of (product of runlengths of 1-bits) and (product of runlengths of 0-bits) in binary representation of n.

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