A206852 -id:A206852 - cp's OEIS frontend

A206853 a(1)=1, for n>1, a(n) is the least number > a(n-1) such that the Hamming distance D(a(n-1), a(n)) = 2.

1, 2, 4, 7, 11, 13, 14, 22, 26, 28, 31, 47, 55, 59, 61, 62, 94, 110, 118, 122, 124, 127, 191, 223, 239, 247, 251, 253, 254, 382, 446, 478, 494, 502, 506, 508, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1534, 1790, 1918, 1982, 2014, 2030, 2038, 2042

Offset: 1

Vladimir Shevelev, Feb 13 2012

For integers a, b, denote by a<+>b the least c>=a, such that D(a,c)=b (note that, generally speaking, a<+>b differs from b<+>a). Then a(n+1)=a(n)<+>2. Thus this sequence is a Hamming analog of odd numbers 1,3,5,...
A Hamming analog of nonnegative integers is A000225 and a Hamming analog of the triangular numbers is A000975.
All terms are odious (A000069).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A182187 (n<+>2), A207063 (starting from 0).

Cf. A000225, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A000069.

read("transforms");
Hamming := proc(a,b)
        XORnos(a,b) ;
        wt(%) ;
end proc:
Dplus := proc(a,b)
        for c from a to 1000000 do
                if Hamming(a,c)=b then
                        return c;
                end if;
        end do:
        return -1 ;
end proc:
A206853 := proc(n)
        option remember;
        if n = 1 then
                1;
        else
                Dplus(procname(n-1),2) ;
        end if;
end proc: # R. J. Mathar, Apr 05 2012

myHammingDistance[n_, m_] := Module[{g = Max[m, n], h = Min[m, n]}, b1 = IntegerDigits[g, 2]; b2 = IntegerDigits[h, 2, Length[b1]]; HammingDistance[b1, b2]]; t = {1}; Do[If[myHammingDistance[t[[-1]], n] == 2, AppendTo[t, n]], {n, 2, 2042}]; t (* T. D. Noe, Mar 07 2012 *)
t={x=1}; Do[i=x+1; While[Count[IntegerDigits[BitXor[x,i],2],1]!=2,i++]; AppendTo[t,x=i],{n,53}]; t (* Jayanta Basu, May 26 2013 *)

next_A206853(n)={my(b=binary(n)); until(norml2(binary(n)-b)==2, n++>=2^#b & b=concat(0,b)); n}
print1(n=1); for(i=1,99,print1(","n=next_A206853(n)))  \\ M. F. Hasler, Apr 07 2012

A206960 a(0)=0, a(1)=1, for n>1, a(n) = n<>n, where k<>m = k<+>k<+>...<+>k is the m-1-fold iteration of the operation <+> defined in A206853.

Original entry on oeis.org

0, 1, 4, 9, 31, 66, 190, 385, 1022, 2055, 5112, 10242, 24572, 49153, 114686, 229377, 524287, 1048590, 2359281, 4718599, 10485753, 20971521, 46137342, 92274689, 201326590, 402653191, 872415224, 1744830466, 3758096380, 7516192769, 16106127358, 32212254721

Offset: 0

Vladimir Shevelev and Konstantin Shukhmin (shukhmin(AT)callsouth.net.nz), Feb 14 2012

A Hamming's analog of n^2.

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853.

f[n_,k_]:=Module[{i=n+1},While[Count[IntegerDigits[BitXor[n,i],2],1]!=k,i++]; i]; Join[{0,1},Table[Nest[f[#,k]&,k,k-1],{k,2,18}]] (* Jayanta Basu, May 26 2013 *)

A010060(a(n)) = A010060(n). - Vladimir Shevelev and Alois P. Heinz, Feb 17 2012

a(11)-a(31) from Alois P. Heinz, Feb 16 2012

A182187 a(n) is the least m >= n such that the Hamming distance D(n,m) = 2.

Original entry on oeis.org

3, 2, 4, 5, 7, 6, 10, 11, 11, 10, 12, 13, 15, 14, 22, 23, 19, 18, 20, 21, 23, 22, 26, 27, 27, 26, 28, 29, 31, 30, 46, 47, 35, 34, 36, 37, 39, 38, 42, 43, 43, 42, 44, 45, 47, 46, 54, 55, 51, 50, 52, 53, 55, 54, 58, 59, 59, 58, 60, 61, 63, 62, 94, 95, 67, 66, 68

Offset: 0

Vladimir Shevelev, Apr 17 2012

a(n) = n<+>2 (see comment in A206853).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A206853 (trajectory of 1), A207063 (trajectory of 0).
Cf. A209544 (primes which are not terms), A209554 (and also not n<+>3).
Cf. A086799 ((n-1)<+>1), A182209 (n<+>3), A182336 (n<+>4).
Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206960, A007814.

HD:= (i, j)-> add(h, h=Bits[Split](Bits[Xor](i, j))):
a:= proc(n) local c;
      for c from n do if HD(n, c)=2 then return c fi od
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 17 2012

t={}; Do[i=n+1; While[Count[IntegerDigits[BitXor[n,i],2],1]!=2,i++]; AppendTo[t,i],{n,0,66}]; t (* Jayanta Basu, May 26 2013 *)

a(n) = bitxor(n, 3<>1+1,2)); \\ Kevin Ryde, Jul 09 2021

def a(n):
  m = n + 1
  while bin(n^m).count('1') != 2: m += 1
  return m
print([a(n) for n in range(67)]) # Michael S. Branicky, Mar 02 2021

def A182187(n):
    S = n.bits(); T = S; c = n; L = len(S)
    while true:
         H = sum(a != b for a, b in zip(S, T))
         if H == 2: return c
         c += 1; T = c.bits()
         if len(T) > L: L += 1; S.append(0)
[A182187(n) for n in (0..66)]   # Peter Luschny, May 26 2013

If n is odd, then a(n) = n+2^(A007814(n+1)-1); if n == 2 (mod 4), then a(n) = n+2^(A007814(n+2)-1); if n == 0 (mod 4), then a(n) = n+3.
Using this formula, we can prove the conjecture formulated in comment in A209554 in the case k=2. Moreover, let us show that if N does not have the form 8*t or 8*t+1, then it can be represented in the form n<+>2. Indeed, in the cases N = 8*t+2, 8*t+4, 8*t+6, 8*t+3, 8*t+5 and 8*t+7 it is sufficient to choose n=N-4, n=N-2, n=N-1, n=N-3, n=N-2 and n = N-3 respectively; in the cases 8*t, 8*t+1, for every choice of n <= N, we do not obtain the equality n<+>2 = N.
In addition, note that n<+>1 = n + 2^A007814(n+1) = A086799(n+1).

More terms from Alois P. Heinz, Apr 17 2012

A182209 a(n) is the least m >= n, such that the Hamming distance D(n,m) = 3.

Original entry on oeis.org

7, 6, 5, 4, 9, 8, 8, 9, 15, 14, 13, 12, 16, 17, 18, 19, 23, 22, 21, 20, 25, 24, 24, 25, 31, 30, 29, 28, 36, 37, 38, 39, 39, 38, 37, 36, 41, 40, 40, 41, 47, 46, 45, 44, 48, 49, 50, 51, 55, 54, 53, 52, 57, 56, 56, 57, 63, 62, 61, 60, 76, 77, 78, 79, 71, 70, 69

Offset: 0

Vladimir Shevelev, Apr 18 2012

a(n) = n<+>3 (see comment in A206853).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A086799 ((n-1)<+>1), A182187 (n<+>2), A182336 (n<+>4).
Cf. A209554 (primes which are not terms nor n<+>2 terms).
Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206960, A007814.

HD:= (i, j)-> add(h, h=Bits[Split](Bits[Xor](i, j))):
a:= proc(n) local c;
      for c from n do if HD(n, c)=3 then return c fi od
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 18 2012

a(n) = bitxor(n, if(bitand(n,14)==4, 13, 7<>2+1,2))); \\ Kevin Ryde, Jul 10 2021

def d(n, m): return bin(n^m).count('1')
def a(n):
    m = n+1
    while d(n, m) != 3: m += 1
    return m
print([a(n) for n in range(67)]) # Michael S. Branicky, Jul 06 2021

If n==i mod 8, then a(n) = n-2*i+7, i=0,1,2,3; if n==4 mod 16, then a(n) = n+5; if n==12 mod 16, then a(n) = n+2^(A007814(n+4)-2); if n==5 mod 16, then a(n) = n+3; if n==13 mod 16, then a(n) = n+2^(A007814(n+3)-2); if n==6 mod 8, then a(n) = n+2^(A007814(n+2)-2); if n==7 mod 8, then a(n) = n+2^(A007814(n+1)-2).

Using this formula, we can prove conjecture formulated in comment in A209554 in case k=3. Moreover, one can prove that N could be represented in form n<+>2 or n<+>3 iff N is not a number of the forms 32*t, 32*t+1. - Vladimir Shevelev, Apr 25 2012

A207016 a(1) = 3 , for n > 1 a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 3.

Original entry on oeis.org

3, 4, 9, 14, 18, 21, 24, 31, 39, 41, 46, 50, 53, 56, 63, 79, 83, 84, 89, 94, 102, 104, 111, 115, 116, 121, 126, 158, 166, 168, 175, 179, 180, 185, 190, 206, 210, 213, 216, 223, 231, 233, 238, 242, 245, 248, 255, 319, 335, 339, 340, 345, 350, 358, 360, 367, 371

Offset: 1

Vladimir Shevelev, Feb 14 2012

Odious and evil terms are alternating (cf. A000069, A001969).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A000069, A000225, A001969, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853.

a[1] = 3; a[n_] := a[n] = Module[{k = a[n - 1], m = a[n-1] + 1}, While[DigitCount[BitXor[k, m], 2, 1] != 3, m++]; m]; Array[a, 100] (* Amiram Eldar, Aug 06 2023 *)

list(n)=my(v=vector(n));v[1]=3;for(i=2,n,v[i]=v[i-1]; while(hammingweight(bitxor(v[i-1],v[i]++))!=3,));v \\ Charles R Greathouse IV, Mar 26 2013

A208982 Numbers n such that the next larger number with mutual Hamming distance 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 16, 17, 18, 19, 21, 22, 23, 27, 28, 29, 30, 36, 39, 40, 41, 42, 43, 45, 46, 52, 57, 58, 60, 63, 65, 66, 67, 69, 70, 71, 72, 75, 77, 78, 81, 82, 88, 95, 96, 99, 100, 101, 102, 105, 106, 108, 111, 112, 119, 123, 125, 126, 129, 130, 136, 137, 138, 147, 148, 149, 150

Offset: 1

Vladimir Shevelev, Mar 04 2012

If p is prime, then p-1 is in the sequence.

Using the prime number theorem in arithmetic progressions k*n+b with gcd(k,b)=1 and its uniformity over k<=exp(c*sqrt(log(x))), one can prove that the counting function of a(n)<=x is equivalent to 2*x/log(x), as x tends to infinity.

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A209085.

isok(n) = my(nextn = n+1); while (hammingweight(bitxor(n, nextn)) != 1, nextn++); isprime(nextn); \\ Michel Marcus, Jul 01 2014

A209544 Primes not expressed in form n<+>2, where operation <+> defined in A206853.

Original entry on oeis.org

3, 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 10 2012

Trivially every odd prime is expressed in form n<+>1 (cf. A208982).

Are these related to A141174, A045390 or A007519? - R. J. Mathar, Mar 13 2012

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A007519, A209554, A182187.

For n>=2, a(n) = A007519(n-1). - Vladimir Shevelev, Apr 18 2012

A182336 a(n) is the least m>=n, such that the Hamming distance D(n,m)=4.

Original entry on oeis.org

15, 14, 13, 12, 11, 10, 9, 8, 19, 18, 17, 16, 17, 16, 16, 17, 31, 30, 29, 28, 27, 26, 25, 24, 33, 32, 32, 33, 32, 33, 34, 35, 47, 46, 45, 44, 43, 42, 41, 40, 51, 50, 49, 48, 49, 48, 48, 49, 63, 62, 61, 60, 59, 58, 57, 56, 64, 65, 66, 67, 68, 69, 70, 71, 79, 78, 77, 76, 75, 74

Offset: 0

Vladimir Shevelev, Apr 25 2012

Or (see comment in A206853) a(n)=n<+>4.

Conjecture: for n > 96, n + 1 <= a(n) <= 9n/8 + 1. - Charles R Greathouse IV, Apr 25 2012

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A207063, A209544, A209554, A007814, A086799, A182187, A182209.

hamming(n)=my(v=binary(n));sum(i=1,#v,v[i])
a(n)=my(k=n);while(hamming(bitxor(n,k++))!=4,);k \\ Charles R Greathouse IV, Apr 25 2012

Terms corrected by Charles R Greathouse IV, Apr 25 2012

A207460 Let a(1) = 4. For n > 1, a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 4.

Original entry on oeis.org

4, 11, 16, 31, 35, 44, 49, 62, 70, 73, 82, 93, 97, 110, 112, 127, 143, 145, 158, 162, 173, 176, 191, 199, 200, 211, 220, 224, 239, 241, 254, 286, 290, 301, 304, 319, 327, 328, 339, 348, 352, 367, 369, 382, 398, 400, 415, 419

Offset: 1

Vladimir Shevelev, Feb 18 2012

All terms are odious (A000069).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000225, A205509, A205510, A205511, A205302, 205649, A205533, A122565, A206852, A206853, A206960, A207016, A207017.

nxt[n_]:=Module[{k=n+1},While[HammingDistance[PadLeft[IntegerDigits[ n,2],IntegerLength[ k,2]],IntegerDigits[k,2]]!=4,k++];k]; NestList[ nxt,4,50] (* Harvey P. Dale, Nov 07 2020 *)

A207472 Let a(1) = 5. For n > 1, a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 5.

Original entry on oeis.org

5, 26, 33, 62, 66, 93, 96, 127, 135, 152, 163, 188, 192, 223, 225, 254, 270, 273, 294, 313, 320, 351, 353, 382, 390, 409, 418, 445, 449, 478, 480, 511, 543, 545, 574, 578, 605, 608, 639, 647, 664, 675, 700, 704, 735, 737, 766, 782, 785, 806, 825, 832, 863, 865

Offset: 1

Vladimir Shevelev, Feb 18 2012

Odious and evil terms are alternating (cf. A000069, A001969).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000069, A000225, A001969, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A207016, A207017.

a[1] = 5; a[n_] := a[n] = Module[{k = a[n - 1], m = a[n-1] + 1}, While[DigitCount[BitXor[k, m], 2, 1] != 5, m++]; m]; Array[a, 100] (* Amiram Eldar, Aug 06 2023 *)

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A206853 a(1)=1, for n>1, a(n) is the least number > a(n-1) such that the Hamming distance D(a(n-1), a(n)) = 2.

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A206960 a(0)=0, a(1)=1, for n>1, a(n) = n<*>n, where k<*>m = k<+>k<+>...<+>k is the m-1-fold iteration of the operation <+> defined in A206853.

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A182187 a(n) is the least m >= n such that the Hamming distance D(n,m) = 2.

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A182209 a(n) is the least m >= n, such that the Hamming distance D(n,m) = 3.

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A207016 a(1) = 3 , for n > 1 a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 3.

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A208982 Numbers n such that the next larger number with mutual Hamming distance 1 is prime.

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A209544 Primes not expressed in form n<+>2, where operation <+> defined in A206853.

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A182336 a(n) is the least m>=n, such that the Hamming distance D(n,m)=4.

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A206960 a(0)=0, a(1)=1, for n>1, a(n) = n<>n, where k<>m = k<+>k<+>...<+>k is the m-1-fold iteration of the operation <+> defined in A206853.