A227190 -id:A227190 - cp's OEIS frontend

A236840 n minus number of runs in the binary expansion of n: a(n) = n - A005811(n).

0, 0, 0, 2, 2, 2, 4, 6, 6, 6, 6, 8, 10, 10, 12, 14, 14, 14, 14, 16, 16, 16, 18, 20, 22, 22, 22, 24, 26, 26, 28, 30, 30, 30, 30, 32, 32, 32, 34, 36, 36, 36, 36, 38, 40, 40, 42, 44, 46, 46, 46, 48, 48, 48, 50, 52, 54, 54, 54, 56, 58, 58, 60, 62, 62, 62, 62, 64, 64, 64

Offset: 0

Antti Karttunen, Apr 18 2014

All terms are even. Used by the "number-of-runs beanstalk" sequence A255056 and many of its associated sequences.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A091067 (the positions of records), A106836 (run lengths).
Cf. A255070 (terms divided by 2).
Cf. A005811, A000120, A003188.
Cf. A255056, A255066, A255061, A255062, A255071, A255072, A255058, A255327.
Other subtracting maps: A011371, A178503, A219641, A219651, A227190, A227191, A237449, A244320, A244234, A255131.

A236840 := proc(n) local i, b; if n=0 then 0 else b := convert(n, base, 2); select(i -> (b[i-1]<>b[i]), [$2..nops(b)]); n-1-nops(%) fi end: seq(A236840(i), i=0..69); # Peter Luschny, Apr 19 2014

a[n_] := n - Length@ Split[IntegerDigits[n, 2]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 16 2023 *)

Scheme
```
(define (A236840 n)  (- n (A005811 n)))
```

a(n) = n - A005811(n) = n - A000120(A003188(n)).

a(n) = 2*A255070(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

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

A227191 a(n) = n minus (product of nonzero digits in factorial base representation of n).

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 12, 13, 15, 16, 17, 18, 16, 17, 23, 24, 25, 26, 26, 27, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 36, 37, 39, 40, 41, 42, 40, 41, 46, 47, 48, 49, 48, 49, 52, 53, 54, 55, 54, 55, 56, 57, 58, 59, 56, 57, 60, 61, 62

Offset: 1

Antti Karttunen, Jul 04 2013

			22 has factorial expansion A007623(22) = "320", and multiplying the nonzero digits, we get 3*2 = 6, and 22-6 = 16, thus a(22)=16.

Antti Karttunen, Table of n, a(n) for n = 1..5040
Index entries for sequences related to factorial base representation.

Cf. A007623, A227153, A219651, A227190.

a[n_] := Module[{k = n, m = 2, r, p = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, If[r > 0, p *= r]; m++]; n - p]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

Scheme
```
(define (A227191 n) (- n (A227153 n)))
```

a(n) = n - A227153(n).

A237449 a(n) = n - A236855(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 4, 5, 5, 5, 7, 7, 7, 7, 13, 13, 14, 14, 14, 17, 17, 18, 18, 18, 20, 20, 20, 20, 25, 25, 26, 26, 26, 28, 28, 28, 28, 31, 31, 31, 31, 31, 41, 41, 42, 42, 42, 45, 45, 46, 46, 46, 48, 48, 48, 48, 54, 54, 55, 55, 55, 58, 58, 59, 59, 59, 61, 61, 61, 61

Offset: 0

Antti Karttunen, Apr 18 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16796

Cf. A236855, A239903, A236859.

Cf. also A011371, A219641, A219651, A227190, A227191, A236840.

A236855list[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Total[r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
With[{m = 6}, Range[0, CatalanNumber[m]-1] - A236855list[m]] (* Generates C(m) terms *) (* Paolo Xausa, Feb 20 2024 *)

Scheme
```
(define (A237449 n) (- n (A236855 n)))
```

a(n) = n - A236855(n).

A283972 a(n) = n minus (product of lengths of 1-runs in binary representation of n) = n - A227349(n).

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 4, 7, 8, 9, 9, 10, 11, 11, 11, 15, 16, 17, 17, 19, 20, 20, 20, 22, 23, 24, 23, 25, 26, 26, 26, 31, 32, 33, 33, 35, 36, 36, 36, 39, 40, 41, 41, 42, 43, 43, 43, 46, 47, 48, 47, 50, 51, 50, 49, 53, 54, 55, 53, 56, 57, 57, 57, 63, 64, 65, 65, 67, 68, 68, 68, 71, 72, 73, 73, 74, 75, 75, 75, 79, 80, 81, 81, 83, 84, 84, 84, 86, 87, 88, 87, 89

Offset: 1

Antti Karttunen, Apr 14 2017

Antti Karttunen, Table of n, a(n) for n = 1..10923
Index entries for sequences related to binary expansion of n

Cf. A227349.

Cf. A227190, A284560, A284581 for similar sequences.

Scheme
```
(define (A283972 n) (- n (A227349 n)))
```

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

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

Antti Karttunen, Apr 14 2017

Antti Karttunen, Table of n, a(n) for n = 1..10922
Index entries for sequences related to binary expansion of n

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.

Scheme
```
(define (A284581 n) (- n (A284579 n)))
```

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

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).

A284560 a(n) = n minus (LCM of run lengths in binary representation of n) = n - A284559(n).

Original entry on oeis.org

0, 1, 1, 2, 4, 4, 4, 5, 7, 9, 9, 10, 11, 11, 11, 12, 14, 16, 17, 18, 20, 20, 20, 18, 23, 24, 25, 22, 26, 26, 26, 27, 29, 31, 29, 34, 35, 36, 33, 37, 39, 41, 41, 42, 43, 43, 43, 44, 43, 48, 49, 50, 51, 52, 49, 53, 51, 55, 53, 56, 57, 57, 57, 58, 60, 62, 63, 62, 66, 64, 68, 66, 71, 72, 73, 74, 75, 72, 75, 76, 78, 80, 81, 82, 84, 84, 84, 82, 87, 88, 89, 86

Offset: 1

Antti Karttunen, Apr 14 2017

Antti Karttunen, Table of n, a(n) for n = 1..10922
Index entries for sequences related to binary expansion of n

Cf. A284559.

Cf. A227190, A283972, A284581 for similar sequences.

Scheme
```
(define (A284560 n) (- n (A284559 n)))
```

a(n) = n - A284559(n).

cp's OEIS Frontend

A236840 n minus number of runs in the binary expansion of n: a(n) = n - A005811(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Scheme

Formula

A167489 Product of run lengths in binary representation of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Scheme

Formula

A227191 a(n) = n minus (product of nonzero digits in factorial base representation of n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A237449 a(n) = n - A236855(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A283972 a(n) = n minus (product of lengths of 1-runs in binary representation of n) = n - A227349(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Scheme

Formula

A284560 a(n) = n minus (LCM of run lengths in binary representation of n) = n - A284559(n).

Views