A062383 -id:A062383 - cp's OEIS frontend

A086099 a(n) = OR(k AND (n-k): 0<=k<=n), AND and OR bitwise.

0, 0, 1, 0, 3, 2, 3, 0, 7, 6, 7, 4, 7, 6, 7, 0, 15, 14, 15, 12, 15, 14, 15, 8, 15, 14, 15, 12, 15, 14, 15, 0, 31, 30, 31, 28, 31, 30, 31, 24, 31, 30, 31, 28, 31, 30, 31, 16, 31, 30, 31, 28, 31, 30, 31, 24, 31, 30, 31, 28, 31, 30, 31, 0, 63, 62, 63, 60, 63, 62, 63, 56, 63, 62

Offset: 0

Reinhard Zumkeller, Jul 09 2003

a(2^n - 1) = 0, a(3*2^n - 1) = 2^n;

A086100(n) = A007088(a(n)).

			a(4) = (0 AND 4) OR (1 AND 3) OR (2 AND 2) OR (3 AND 1) OR (4 AND 0) -> (000 AND 100) OR (001 AND 011) OR (010 AND 010) OR (011 AND 001) OR (111 AND 000) = 000 OR 011 OR 010 OR 011 OR 000 = 011 -> a(4)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, AND
Eric Weisstein's World of Mathematics, OR
Reinhard Zumkeller, Logical Convolutions

Cf. A003817 (even bisection), A062383.
Cf. A086100 (in binary), A007088.
Cf. A053644, A006519.
Cf. A000004, A142149, A142150, A142151, A001477.

import Data.Bits ((.&.), (.|.))
a086099 n = foldl1 (.|.) $ zipWith (.&.) [0..] $ reverse [0..n] :: Integer
-- Reinhard Zumkeller, Jun 04 2012

a[n_] := BitOr @@ Table[BitAnd[k, n - k], {k, 0, n}]; Table[a[n], {n, 0, 73}] (* Jean-François Alcover, Jun 19 2012 *)

PARI
```
a(n) = n++; 1<Kevin Ryde, Apr 11 2023
```

a(2*n) = 2*2^floor(log_2(n)) - 1 = A003817(n).
a(2*n+1) = 2*a(n).
a(n) = A053644(n+1) - A006519(n+1). - Ridouane Oudra, Apr 09 2023

A261644 Distance of A260273(n) to next power of 2.

Original entry on oeis.org

1, 1, 3, 8, 5, 1, 15, 12, 9, 5, 1, 31, 28, 25, 20, 13, 8, 3, 63, 60, 57, 52, 47, 44, 41, 37, 33, 29, 24, 17, 13, 8, 3, 127, 124, 121, 116, 111, 108, 105, 99, 91, 88, 85, 81, 77, 70, 66, 62, 57, 52, 47, 40, 33, 29, 24, 15, 10, 6, 2, 254, 251, 248, 245, 239

Offset: 1

Reinhard Zumkeller, Aug 30 2015

This sequence, as well as A261712, is suggested by A261396 and A261416.

			.  1: 1
.  2: 1
.  3: 3
.  4: 8,5,1
.  5: 15,12,9,5,1
.  6: 31,28,25,20,13,8,3
.  7: 63,60,57,52,47,44,41,37,33,29,24,17,13,8,3
.  8: 127,124,121,116,111,108,105,99,91,88,85,81,77,70,... (27 terms)
.  9: 254,251,248,245,239,236,233,227,218,213,207,202,195,,... (49 terms)

Reinhard Zumkeller, Rows n = 1..20 of triangle, flattened

Cf. A260273, A062383, A261645 (first differences), A261712 (reversed), A261646 (row lengths).

a261644 n = a261644_list !! (n-1)
a261644_list = zipWith (-)
               (map a062383 a260273_list) $ map fromIntegral a260273_list
a261644_tabf = [1] : f (tail $ zip a261645_list a261644_list) where
   f dxs = (map snd (dxs'' ++ [dx])) : f dxs' where
     (dxs'', dx:dxs') = span ((<= 0) . fst) dxs
a261644_row n = a261644_tabf !! (n-1)

a(n) = A062383(A260273(n)) - A260273(n).

A385552 Period of {binomial(N,n) mod 5: N in Z}.

Original entry on oeis.org

1, 5, 5, 5, 5, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125

Offset: 0

Jianing Song, Jul 03 2025

a(n) is the smallest power of 5 > n. For the general result, see A349593.

Since the modulus (5) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem.

			For N == 0, 1, ..., 24 (mod 5), binomial(N,5) == {0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4} (mod 5).

Jianing Song, Table of n, a(n) for n = 0..625
Wikipedia, Lucas's theorem

Column 5 of A349593. A062383, A064235 (if offset 0), A385553, and A385554 are respectively columns 2, 3, 6, and 10.

PARI
```
a(n) = if(n, 5^(logint(n,5)+1), 1)
```

from sympy import integer_log
def A385552(n): return 5*5**(integer_log(n,5)[0]) if n else 1 # Chai Wah Wu, Jul 06 2025

A385553 Period of {binomial(N,n) mod 6: N in Z}.

Original entry on oeis.org

1, 6, 12, 36, 72, 72, 72, 72, 144, 432, 432, 432, 432, 432, 432, 432, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 2592, 2592, 2592, 2592, 2592, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184

Offset: 0

Jianing Song, Jul 03 2025

			For N == 0, 1, ..., 35 (mod 36), binomial(N,3) == {0, 0, 0, 1, 4, 4, 2, 5, 2, 0, 0, 3, 4, 4, 4, 5, 2, 2, 0, 3, 0, 4, 4, 1, 2, 2, 2, 3, 0, 0, 4, 1, 4, 2, 2, 5} (mod 6).
For N == 0, 1, ..., 71 (mod 72), binomial(N,4) == {0, 0, 0, 0, 1, 5, 3, 5, 4, 0, 0, 0, 3, 1, 5, 3, 2, 4, 0, 0, 3, 3, 1, 5, 0, 2, 4, 0, 3, 3, 3, 1, 2, 0, 2, 4, 3, 3, 3, 3, 4, 2, 0, 2, 1, 3, 3, 3, 0, 4, 2, 0, 5, 1, 3, 3, 0, 0, 4, 2, 3, 5, 1, 3, 0, 0, 0, 4, 5, 3, 5, 1} (mod 6).

Jianing Song, Table of n, a(n) for n = 0..1024

Column 6 of A349593. A062383, A064235 (if offset 0), A385552, and A385554 are respectively columns 2, 3, 5, and 10.

a(n) = if(n, (2^(logint(n,2)+1)) * (3^(logint(n,3)+1)), 1)

a(n) = (the smallest power of 2 > n) * (the smallest power of 3 > n) = A062383(n) * A064235(n+1). For the general result, see A349593.

A385554 Period of {binomial(N,n) mod 10: N in Z}.

Original entry on oeis.org

1, 10, 20, 20, 40, 200, 200, 200, 400, 400, 400, 400, 400, 400, 400, 400, 800, 800, 800, 800, 800, 800, 800, 800, 800, 4000, 4000, 4000, 4000, 4000, 4000, 4000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000

Offset: 0

Jianing Song, Jul 03 2025

			For N == 0, 1, ..., 19 (mod 20), binomial(N,3) == {0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9} (mod 10).
For N == 0, 1, ..., 39 (mod 40), binomial(N,4) == {0, 0, 0, 0, 1, 5, 5, 5, 0, 6, 0, 0, 5, 5, 1, 5, 0, 0, 0, 6, 5, 5, 5, 5, 6, 0, 0, 0, 5, 1, 5, 5, 0, 0, 6, 0, 5, 5, 5, 1} (mod 10).

Jianing Song, Table of n, a(n) for n = 0..1024

Column 10 of A349593. A062383, A064235 (if offset 0), A385552, and A385553 are respectively columns 2, 3, 5, and 6.

a(n) = if(n, (2^(logint(n,2)+1)) * (5^(logint(n,5)+1)), 1)

a(n) = (the smallest power of 2 > n) * (the smallest power of 5 > n) = A062383(n) * A385552(n). For the general result, see A349593.

A083402 Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the main diagonal and 1 elsewhere; a(n) is the size of the conjugacy class of this matrix in GL(n,2).

Original entry on oeis.org

1, 3, 42, 2520, 624960, 629959680, 2560156139520, 41781748196966400, 2732860586067178291200, 715703393163961188325785600, 750102961052993818881476159078400, 3145391744524297920839316348340273152000, 52764474940208177704130232748554603290689536000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jun 12 2003

nonn

			For example for n=4 the matrix is / 1,1,1,1 / 0,1,1,1 / 0,0,1,1 / 0,0,0,1 /.

Eric M. Schmidt, Table of n, a(n) for n = 1..30

Cf. A002884, A070731, A082877, A062383.

a:= n-> 2^((n-1)*(n-2)/2) *mul(2^k-1, k=1..n):
seq(a(n), n=1..15);  # Alois P. Heinz, May 14 2013

a[n_] := 2^((n-1)*(n-2)/2)*Product[2^k-1, {k, 1, n}]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

a(n) = A002884(n) / 2^(n-1).

More terms from Eric M. Schmidt, May 14 2013

A105670 a(1)=1 then bracketing n by powers of 2 as f(t)=2^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 5, 5, 15, 15, 13, 13, 9, 9, 11, 11, 31, 31, 29, 29, 25, 25, 27, 27, 17, 17, 19, 19, 23, 23, 21, 21, 63, 63, 61, 61, 57, 57, 59, 59, 49, 49, 51, 51, 55, 55, 53, 53, 33, 33, 35, 35, 39, 39, 37, 37, 47, 47, 45, 45, 41, 41, 43, 43, 127, 127, 125, 125, 121, 121, 123

Offset: 1

Benoit Cloitre, May 03 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A006068, A105669, A105672, A093347, A093348.

A062383 := proc(n)
        ceil(log(n)/log(2)) ;
        2^% ;
end proc:
A105670 := proc(n)
        option remember;
        if n = 1 then
                1;
        else
                fn1 := A062383(n) ;
                fn := fn1/2 ;
                fn1-procname(n-fn) ;
        end if;
end proc:
seq(A105670(n),n=1..80) ; # R. J. Mathar, Nov 06 2011

t[0] = 0; t[1] = 1; t[n_?EvenQ] := t[n] = t[n/2]; t[n_?OddQ] := t[n] = 1 - t[(n-1)/2]; a[1] = 1; a[n_?EvenQ] := a[n] = a[n - 1]; a[n_] := a[n] = 2*a[Ceiling[n/2]] - 1 + 2*t[Ceiling[n/2] - 1]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Aug 13 2013 *)

b(n,m)=if(n<2,1,m*m^floor(log(n-1)/log(m))-b(n-m^floor(log(n-1)/log(m)),m))

a(2n-1) = a(2n).
a(n) = 2*a(ceiling(n/2)) -1 + 2*t(ceiling(n/2)-1) where t(n) = A010060(n) is the Thue-Morse sequence.
a(2n-1) = a(2n) = 2*A006068(n-1)+1. - Jeffrey Shallit, Mar 15 2025

Typo in data corrected by Jean-François Alcover, Aug 13 2013

A295609 a(n) = least prime number p such that p AND n = n (where AND denotes the binary AND operator).

Original entry on oeis.org

2, 3, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 31, 31, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 31, 31, 29, 29, 31, 31, 37, 37, 43, 43, 37, 37, 47, 47, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 59, 59, 53, 53, 127, 127, 59, 59, 59, 59, 61, 61, 127, 127, 67, 67

Offset: 0

Rémy Sigrist, Nov 24 2017

For any n > 0: gcd(A109613(n), A062383(n)) = 1, hence, by Dirichlet's theorem on arithmetic progressions, we have a prime number, say p, of the form A109613(n) + k * A062383(n) with k > 0; this prime number satisfies p AND n = n; also a(0) = 2, hence the sequence is well defined for any n >= 0.
a(n) = n iff n is prime.
Each prime number appears 2*k times in this sequence for some k > 0.

			a(42) = 42 + A295335(42) = 42 + 1 = 43.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the first 2^17 terms

Cf. A062383, A109613, A295335.

Table[Block[{p = 2}, While[BitAnd[p, n] != n, p = NextPrime@ p]; p], {n, 0, 65}] (* Michael De Vlieger, Nov 26 2017 *)

avoid(n,i) = if (i, if (n%2, 2*avoid(n\2,i), 2*avoid(n\2,i\2)+(i%2)), 0) \\ (i+1)-th number k such that k AND n = 0
a(n) = for (i=0, oo, my (k=avoid(n,i)); if (isprime(n+k), return (n+k)))

a(n) = n + A295335(n).

For any k > 1, a(2*k) = a(2*k+1).

A065159 Binary string self-substitutions: a(n) is obtained by substituting the binary expansion of n for each 1-bit in the binary expansion of n.

Original entry on oeis.org

0, 1, 4, 15, 16, 85, 108, 511, 64, 585, 660, 5819, 816, 7085, 7644, 65535, 256, 4369, 4644, 78451, 5200, 87381, 91564, 1531639, 6336, 105625, 109876, 1825659, 118384, 1961821, 2029500, 33554431, 1024, 33825, 34884, 1149155, 37008, 1217189, 1250124, 41056743

Offset: 0

Marc LeBrun, Oct 18 2001

			a(5): 5 = 101 -> (101)0(101) = 1010101 = 85.

Paul Tek, Table of n, a(n) for n = 0..10000

Cf. A007088, A065157, A065158, A065160.

bss[n_]:=Module[{idn2=IntegerDigits[n,2]},FromDigits[Flatten[idn2/.{1-> idn2}],2]]; Array[bss,40,0] (* Harvey P. Dale, Aug 15 2017 *)

def a(n): b = bin(n)[2:]; return int(b.replace("1", b), 2)
print([a(n) for n in range(40)]) # Michael S. Branicky, Aug 05 2022

a(0) = 0. a(2^n) = 4^n. a(4n+2) = (4n+2)*(1+a(4n+1)/(4n+1)).
a(n) = A065157(n,n) = A065158(n,n)*n = A065160(n)*n.
a(n) =z(n, n) with z(u, v) = if u=0 then 0 else if u mod 2 = 0 then z(u/2, v)*2 else z([u/2], v)*A062383(v)+v. - Reinhard Zumkeller, Feb 15 2004

Name clarified by Michael S. Branicky, Aug 05 2022

A179857 Smallest number greater than n having in binary representation exactly twice the number of ones as n has in binary representation.

Original entry on oeis.org

3, 3, 15, 5, 15, 15, 63, 9, 15, 15, 63, 15, 63, 63, 255, 17, 23, 23, 63, 23, 63, 63, 255, 27, 63, 63, 255, 63, 255, 255, 1023, 33, 39, 39, 63, 39, 63, 63, 255, 43, 63, 63, 255, 63, 255, 255, 1023, 51, 63, 63, 255, 63, 255, 255, 1023, 63, 255, 255, 1023, 255, 1023, 1023

Offset: 1

Reinhard Zumkeller, Jul 31 2010

a(n) = Min{m: m > n and A000120(m) = 2*A000120(n)};
a(n) is odd;
n < a(n) < A000290(A062383(n));
a(A000079(n)) = A000051(n);
A024036 and A000225 give record values and where they occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to binary expansion of n

Cf. A000120, A086799.
Cf. A000290, A062383.
Cf. A000051, A000079.
Cf. A000225, A024036.

br2[n_]:=Module[{k=If[EvenQ[n],n+1,n+2],t=2*DigitCount[n,2,1]},While[ DigitCount[ k,2,1]!=t,k=k+2];k]; Array[br2,70] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = my(k=n+1, h=hammingweight(n)); while (hammingweight(k) != 2*h, k++); k; \\ Michel Marcus, Nov 13 2023

Definition clarified by Harvey P. Dale, Sep 20 2016

cp's OEIS Frontend

A086099 a(n) = OR(k AND (n-k): 0<=k<=n), AND and OR bitwise.

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A261644 Distance of A260273(n) to next power of 2.

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A385552 Period of {binomial(N,n) mod 5: N in Z}.

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A385553 Period of {binomial(N,n) mod 6: N in Z}.

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A385554 Period of {binomial(N,n) mod 10: N in Z}.

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A083402 Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the main diagonal and 1 elsewhere; a(n) is the size of the conjugacy class of this matrix in GL(n,2).

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A105670 a(1)=1 then bracketing n by powers of 2 as f(t)=2^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).

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