A227191 -id:A227191 - 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).

A227153 Product of nonzero digits of n in factorial base.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 3, 3, 3, 3, 6, 6, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 3, 3, 3, 3, 6, 6, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 8, 8, 6, 6, 6, 6, 12, 12, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Jul 04 2013

a(0) = 1 as an empty product always gives 1.

Antti Karttunen, Table of n, a(n) for n = 0..5040

A227157 gives the positions where equal with A208575.

Cf. A007623, A227154, A227191.

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++]; p]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

from functools import reduce
from operator import mul
def A(n, p=2):
    return n if nIndranil Ghosh, Jun 19 2017

def a(n, k=2): return max(n % k, 1) * a(n // k, k + 1) if n else 1 # David Radcliffe, May 22 2025

For all n, a(A227157(n)) = A208575(A227157(n)).

A227190 a(n) = n minus (product of run lengths in binary representation of 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, 47, 51, 55, 53, 52, 57, 57, 57, 58, 60, 62, 59, 62

Offset: 1

Antti Karttunen, Jul 04 2013

			For 8, "1000" in binary, the run lengths are 1 and 3, 1*3=3, and 8-3 = 5, thus a(8)=5. For 24, "11000" in binary, the run lengths are 2 and 3, 2*3=6, and 24-6 = 18, thus a(24)=18.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A227191, A000975, A084639.

a227190 n = n - a167489 n  -- Reinhard Zumkeller, Jul 05 2013

Table[n-Times@@(Length/@Split[IntegerDigits[n,2]]),{n,70}] (* Harvey P. Dale, Aug 02 2013 *)

(define (A227190 n) (- n (A167489 n))) ;; The Scheme-program for A167489 is found under that entry.

a(n) = n - A167489(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).

cp's OEIS Frontend

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

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A227153 Product of nonzero digits of n in factorial base.

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A227190 a(n) = n minus (product of run lengths in binary representation of n).

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

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