A061601 -id:A061601 - cp's OEIS frontend

A035327 Write n in binary, interchange 0's and 1's, convert back to decimal.

1, 0, 1, 0, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46

Offset: 0

N. J. A. Sloane

For n>0: largest m<=n such that no carry occurs when adding m to n in binary arithmetic: A003817(n+1) = a(n) + n = a(n) XOR n. - Reinhard Zumkeller, Nov 14 2009
a(0) could be considered to be 0 (it was set so from 2004 to 2008) if the binary representation of zero was chosen to be the empty string. - Jason Kimberley, Sep 19 2011
For n > 0: A240857(n,a(n)) = 0. - Reinhard Zumkeller, Apr 14 2014
This is a base-2 analog of A048379. Another variant, without converting back to decimal, is given in A256078. - M. F. Hasler, Mar 22 2015
For n >= 2, a(n) is the least nonnegative k that must be added to n+1 to make a power of 2. Hence in a single-elimination tennis tournament with n entrants, a(n-1) is the number of players given a bye in round one, so that the number of players remaining at the start of round two is a power of 2. For example, if 39 players register, a(38)=25 players receive a round-one bye leaving 14 to play, so that round two will have 25+(14/2)=32 players. - Mathew Englander, Jan 20 2024

			8 = 1000 -> 0111 = 111 = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n
Index entries for sequences related to the Josephus Problem

a(n) = A003817(n) - n, for n>0.
Cf. A080079, A087734, A167831, A167877, A007088, A061601, A171960, A010078, A000225, A006257 (Josephus problem).
Cf. A240857.
Cf. A048379, A256078.

a035327 n = if n <= 1 then 1 - n else 2 * a035327 n' + 1 - b
            where (n',b) = divMod n 2
-- Reinhard Zumkeller, Feb 21 2014

using IntegerSequences
A035327List(len) = [Bits("NAND", n, n) for n in 0:len]
println(A035327List(100))  # Peter Luschny, Sep 25 2021

A035327:=func; // Jason Kimberley, Sep 19 2011

seq(2^(1 + ilog2(max(n, 1))) - 1 - n, n = 0..81); # Emeric Deutsch, Oct 19 2008
A035327 := n -> `if`(n=0, 1, Bits:-Nand(n, n)):
seq(A035327(n), n=0..81); # Peter Luschny, Sep 23 2019

Table[BaseForm[FromDigits[(IntegerDigits[i, 2]/.{0->1, 1->0}), 2], 10], {i, 0, 90}]
Table[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1], {n, 100}] (* Alonso del Arte, Jan 14 2006 *)
Join[{1},Table[2^BitLength[n]-n-1,{n,100}]] (* Paolo Xausa, Oct 13 2023 *)
Table[FromDigits[IntegerDigits[n,2]/.{0->1,1->0},2],{n,0,90}] (* Harvey P. Dale, May 03 2025 *)

a(n)=sum(k=1,n,if(bitxor(n,k)>n,1,0)) \\ Paul D. Hanna, Jan 21 2006

a(n) = bitxor(n, 2^(1+logint(max(n,1), 2))-1) \\ Rémy Sigrist, Jan 04 2019

a(n)=if(n, bitneg(n, exponent(n)+1), 1) \\ Charles R Greathouse IV, Apr 13 2020

def a(n): return int(''.join('1' if i == '0' else '0' for i in bin(n)[2:]), 2) # Indranil Ghosh, Apr 29 2017

def a(n): return 1 if n == 0 else n^((1 << n.bit_length()) - 1)
print([a(n) for n in range(100)]) # Michael S. Branicky, Sep 28 2021

def A035327(n): return (~n)^(-1<Chai Wah Wu, Dec 20 2022

def a(n):
    if n == 0:
        return 1
    return sum([(1 - b) << s for (s, b) in enumerate(n.bits())])
[a(n) for n in srange(82)]  # Peter Luschny, Aug 31 2019

a(n) = 2^k - n - 1, where 2^(k-1) <= n < 2^k.
a(n+1) = (a(n)+n) mod (n+1); a(0) = 1. - Reinhard Zumkeller, Jul 22 2002
G.f.: 1 + 1/(1-x)*Sum_{k>=0} 2^k*x^2^(k+1)/(1+x^2^k). - Ralf Stephan, May 06 2003
a(0) = 0, a(2n+1) = 2*a(n), a(2n) = 2*a(n) + 1. - Philippe Deléham, Feb 29 2004
a(n) = number of positive integers k < n such that n XOR k > n. a(n) = n - A006257(n). - Paul D. Hanna, Jan 21 2006
a(n) = 2^{1+floor(log[2](n))}-n-1 for n>=1; a(0)=1. - Emeric Deutsch, Oct 19 2008
a(n) = if n<2 then 1 - n else 2*a(floor(n/2)) + 1 - n mod 2. - Reinhard Zumkeller, Jan 20 2010
a(n) = abs(2*A053644(n) - n - 1). - Mathew Englander, Jan 22 2024

More terms from Vit Planocka (planocka(AT)mistral.cz), Feb 01 2003
a(0) corrected by Paolo P. Lava, Oct 22 2007
Definition completed by M. F. Hasler, Mar 22 2015

A054054 Smallest digit of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 7, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 0, 0, 0, 0

Offset: 0

Henry Bottomley, Apr 29 2000

a(n) = 0 for almost all n. - Charles R Greathouse IV, Oct 02 2013

More precisely, a(n) = 0 asymptotically almost surely, i.e., except for a set of density 0: As the number of digits of n grows, the probability of having no zero digit goes to zero as 0.9^(length of n), although there are infinitely many counterexamples. - M. F. Hasler, Oct 11 2015

			a(12) = 1 because 1 < 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A054055.

Cf. A061601, A011540, A052382, A262188.

a054054 = f 9 where
   f m x | x <= 9 = min m x
         | otherwise = f (min m d) x' where (x',d) = divMod x 10
-- Reinhard Zumkeller, Jun 20 2012, Apr 25 2012

seq(min(convert(n,base,10)),n=0..100); # Robert Israel, Jul 07 2016

Mathematica
```
A054054[n_]:=Min[IntegerDigits[n]]
```

A054054(n)=if(n,vecmin(digits(n)))  \\ or: Set(digits(n))[1]. - M. F. Hasler, Jan 23 2013

a(A011540(n)) = 0; a(A052382(n)) > 0. - Reinhard Zumkeller, Apr 25 2012
a(n) = A262188(n,0). - Reinhard Zumkeller, Sep 14 2015
a(n) = 0 iff A007954(n) = 0. - M. F. Hasler, Oct 11 2015
a(n) = 9 - A054055(A061601(n)). - Robert Israel, Jul 07 2016

Edited by M. F. Hasler, Oct 11 2015

A055120 Digital complement of n (replace each nonzero digit d with 10-d).

Original entry on oeis.org

0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 90, 99, 98, 97, 96, 95, 94, 93, 92, 91, 80, 89, 88, 87, 86, 85, 84, 83, 82, 81, 70, 79, 78, 77, 76, 75, 74, 73, 72, 71, 60, 69, 68, 67, 66, 65, 64, 63, 62, 61, 50, 59, 58, 57, 56, 55, 54, 53, 52, 51, 40, 49, 48, 47, 46, 45, 44, 43, 42, 41, 30, 39

Offset: 0

Henry Bottomley, Apr 19 2000

a(n) = -n in carryless arithmetic mod 10 - that is, n + a(n) = 0 (cf. A169894). - N. J. A. Sloane, Aug 03 2010

			a(11) = 99 because 1 + 9 = 0 mod 10 for each digit.
a(20) = 80 because 2 + 8 = 0 mod 10 and 0 + 0 = 0 mod 10.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A048647, A055115-A055126, A061601.

Column k=10 of A248813.

a055120 = foldl f 0 . reverse . unfoldr g where
   f v d = if d == 0 then 10 * v else 10 * v + 10 - d
   g x = if x == 0 then Nothing else Just $ swap $ divMod x 10
-- Reinhard Zumkeller, Oct 04 2011

f:=proc(n) local t0,t1,i;
t0:=0; t1:=convert(n,base,10);
for i from 1 to nops(t1) do
if t1[i]>0 then t0:=t0+(10-t1[i])*10^(i-1); fi;
od:
RETURN(t0);
end;
# N. J. A. Sloane, Jan 21 2011

a[n_] := FromDigits[ IntegerDigits[n] /. d_?Positive -> 10-d]; Table[a[n], {n, 0, 100}](* Jean-François Alcover, Nov 28 2011 *)

a(n)=fromdigits(apply(d->if(d,10-d,0),digits(n))) \\ Charles R Greathouse IV, Feb 08 2017

def A055120(n): return int(''.join(str(10-int(d)) if d != '0' else d for d in str(n))) # Chai Wah Wu, Apr 03 2021

From Robert Israel, Sep 04 2017: (Start)
a(10*n) = 10*a(n).
a(10*n+j) = 10*a(n) + 10 - j for 1 <= j <= 9.
G.f. g(x) satisfies g(x) = 10*(1+x+x^2+...+x^9)*g(x^10) + (9*x+8*x^2+7*x^3+6*x^4+5*x^5+4*x^6+3*x^7+2*x^8+x^9)/(1-x^10).
(End)

A135601 Acute-angled numbers with an internal digit as the vertex.

Original entry on oeis.org

102, 103, 104, 105, 106, 107, 108, 109, 120, 130, 131, 132, 140, 141, 142, 143, 150, 151, 152, 153, 154, 160, 161, 162, 163, 164, 165, 170, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 186, 187, 190, 191, 192, 193, 194, 195

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents an acute angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite. The final term has 14 digits: 98765432102468.

			Illustration using the final term of this sequence:
  9 . . . . . . . . . . . . .
  . 8 . . . . . . . . . . . 8
  . . 7 . . . . . . . . . . .
  . . . 6 . . . . . . . . 6 .
  . . . . 5 . . . . . . . . .
  . . . . . 4 . . . . . 4 . .
  . . . . . . 3 . . . . . . .
  . . . . . . . 2 . . 2 . . .
  . . . . . . . . 1 . . . . .
  . . . . . . . . . 0 . . . .

Micah Manary, Table of n, a(n) for n = 1..2337
Micah Manary, Python program

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135602, A135603.

progressions = set(tuple(range(i, j+1, d)) for i in range(10) for d in range(1, 10-i) for j in range(i+d, 10))
s = set()
for left in progressions:
    for right in progressions:
        dl, dr = left[1] - left[0], right[1] - right[0]
        if dl + dr > 2:
            if left[-1] == right[-1]: s.add(left[:-1] + right[::-1])
            if left[0] == right[0]: s.add(left[::-1] + right[1:])
afull = sorted(int("".join(map(str, t))) for t in s if t[0] != 0)
print(afull[:53]) # Michael S. Branicky, Aug 02 2022

If a(n) does not end in 0, then A004086(a(n)) is a term; if a(n) does not start with 9, then A061601(a(n)) is a term. - Michael S. Branicky, Aug 02 2022

A267193 Complement obverse of n.

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 98, 88, 78, 68, 58, 48, 38, 28, 18, 8, 97, 87, 77, 67, 57, 47, 37, 27, 17, 7, 96, 86, 76, 66, 56, 46, 36, 26, 16, 6, 95, 85, 75, 65, 55, 45, 35, 25, 15, 5, 94, 84, 74, 64, 54, 44, 34, 24, 14, 4, 93, 83, 73, 63, 53, 43, 33, 23, 13

Offset: 0

N. J. A. Sloane, Jan 24 2016

Replace digits of n by their 9's complements, reverse the order, omit any leading zeros.

Suggested by a posting by Mike Stay to the Math Fun Mailing List, Jan 24 2016, which in turn was loosely based on the definition used in logic (see the Philosophy Pages link).

			The 9's complement of the digits of 980 are 019, so a(980) = 910.

Alois P. Heinz, Table of n, a(n) for n = 0..9999
Garth Kemerling, Philosophy Pages: Immediate Inferences.

Cf. A061601 (9's complement of n).

a:= n-> (s-> parse(cat(seq(9-s[i], i=1..nops(s))))
         )(convert(n, base, 10)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 24 2016

More terms from Alois P. Heinz, Jan 24 2016

A171960 Values of the 2-complement of n in ternary representation.

Original entry on oeis.org

2, 1, 0, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2

Offset: 0

Reinhard Zumkeller, Jan 20 2010

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A003462, A007089, A035327, A061601, A134025, A134026.

a[n_] := 3^(1 + Floor[Log[3, n]]) - n - 1; a[0] = 2; Array[a, 100] (* Amiram Eldar, Apr 03 2025 *)

a(n) = 3^(if(n,logint(n,3))+1) - 1 - n; \\ Kevin Ryde, Jul 16 2020

a(n) = if n < 3 then 2 - n else 3*a(floor(n/3)) + 2 - n mod 3.
a(A134026(n)) < A134026(n).
a(A003462(n)) = A003462(n).
a(A134025(n)) >= A134025(n).

A109002 Maximal difference between two n-digit numbers.

Original entry on oeis.org

9, 89, 899, 8999, 89999, 899999, 8999999, 89999999, 899999999, 8999999999, 89999999999, 899999999999, 8999999999999, 89999999999999, 899999999999999, 8999999999999999, 89999999999999999, 899999999999999999, 8999999999999999999, 89999999999999999999, 899999999999999999999

Offset: 1

Amarnath Murthy, Aug 14 2005

			a(1) = 9 - 0 = 9, a(4) = 9999 - 1000 = 8999.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A061601, A178500.

[9] cat [9*10^n-1: n in [1..30]]; // Vincenzo Librandi, Oct 29 2011

Join[{9},Table[FromDigits[PadRight[{8},n,9]],{n,2,20}]] (* or *) LinearRecurrence[{11,-10},{9,89,899},20] (* Harvey P. Dale, May 09 2021 *)

a(n)=if(n>1,(10^n-1)-10^(n-1),9) \\ Charles R Greathouse IV, Oct 29 2011

a(n) = (10^n - 1) - 10^(n-1), n > 1.
From Reinhard Zumkeller, May 28 2010: (Start)
a(n) = A061601(A178500(n-1)).
a(n+1) = 10*a(n) + 9. (End)
G.f.: 9*x - x^2*(-89+80*x)/((10*x-1)*(x-1)). - R. J. Mathar, Oct 29 2011
From Elmo R. Oliveira, Jun 12 2025: (Start)
E.g.f.: (1 + 10*x - 10*exp(x) - exp(10*x) + 10*exp(10*x))/10.
a(n) = 11*a(n-1) - 10*a(n-2) for n >= 4. (End)

A178500 a(n) = 10^n * signum(n).

Original entry on oeis.org

0, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000, 100000000000000000000

Offset: 0

Reinhard Zumkeller, May 28 2010

a(n-1) is the minimum difference between an n-digit number (written in base 10, nonzero leading digit) and the product of its digits. For n > 1, it is also a number meeting that bound. See A070565. - Devin Akman, Apr 17 2019

Michael De Vlieger, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A000533, A011557, A057427, A061601, A070565, A109002, A155559, A178501.

Partial sums of A063945.

Array[10^#*Sign[#] &, 20, 0] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = A011557(n)*A057427(n).
For n > 0, a(n) = A011557(n).
a(n) = 10*A178501(n).
a(n) = A000533(n) - 1.
A061601(a(n)) = A109002(n+1).
From Elmo R. Oliveira, Jul 21 2025: (Start)
G.f.: 10*x/(1-10*x).
E.g.f.: 2*exp(5*x)*sinh(5*x).
a(n) = 10*a(n-1) for n > 1. (End)

A084021 Product of n and its 9's complement.

Original entry on oeis.org

0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 890, 968, 1044, 1118, 1190, 1260, 1328, 1394, 1458, 1520, 1580, 1638, 1694, 1748, 1800, 1850, 1898, 1944, 1988, 2030, 2070, 2108, 2144, 2178, 2210, 2240, 2268, 2294, 2318, 2340, 2360, 2378, 2394, 2408, 2420, 2430, 2438

Offset: 0

Amarnath Murthy, May 23 2003

There are infinitely many nonzero squares in this sequence; for example a(10^(4n) - 10^(2n)) = (10^n*(10^(2n)-1))^2. - David Wasserman, Dec 07 2004

Seiichi Manyama, Table of n, a(n) for n = 0..9999

Cf. A061601.

{a(n) = n*(10^length(Str(n))-1-n)} \\ Seiichi Manyama, Sep 15 2019

a(n) = n*(10^d - 1 - n), where d is the number of digits in n.

a(n) = n*A061601(n). - David Wasserman, Dec 07 2004

More terms from David Wasserman, Dec 07 2004

a(0)=0 prepended by Seiichi Manyama, Sep 15 2019

A160668 Distance between prime(n) and the next higher power of 10.

Original entry on oeis.org

8, 7, 5, 3, 89, 87, 83, 81, 77, 71, 69, 63, 59, 57, 53, 47, 41, 39, 33, 29, 27, 21, 17, 11, 3, 899, 897, 893, 891, 887, 873, 869, 863, 861, 851, 849, 843, 837, 833, 827, 821, 819, 809, 807, 803, 801, 789, 777, 773, 771, 767, 761, 759, 749, 743, 737, 731, 729, 723

Offset: 1

Enoch Haga, May 23 2009

			a(1)=8 because 10^1=10, and 10-2, 1st prime 2, = 8;
a(5)=89 because 10^2=100 and 100-11, 5th prime 11, = 89.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A055642, A061601, A160669, A160670, A228628.

[10^(Ceiling(Log(NthPrime(n))/Log(10))) - NthPrime(n): n in [1..30]]; // G. C. Greubel, May 02 2018

a:= n-> (p-> 10^length(p)-p)(ithprime(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 08 2017

Table[10^Ceiling[Log[Prime[n]]/Log[10]] - Prime[n], {n, 1, 100}] (* G. C. Greubel, May 02 2018 *)

a(n) = 10^ceil(log(prime(n))/log(10)) - prime(n); \\ Michel Marcus, Dec 08 2017

20 N=3:print N:C=2
30 A=3:S=sqrt(N)
40 B=N/A 50 if A*B=int(N) then 70
60 A=A+2:if AEnoch Haga, May 22 2009

Formula

From Alois P. Heinz, Dec 08 2017: (Start)

a(n) = 10^A055642(A000040(n)) - A000040(n).

a(n) = A228628(n) + 1 = A061601(A000040(n)) + 1. (End)

cp's OEIS Frontend

A035327 Write n in binary, interchange 0's and 1's, convert back to decimal.

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A054054 Smallest digit of n.

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A055120 Digital complement of n (replace each nonzero digit d with 10-d).

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A135601 Acute-angled numbers with an internal digit as the vertex.

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A267193 Complement obverse of n.

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A171960 Values of the 2-complement of n in ternary representation.

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