A126084 -id:A126084 - cp's OEIS frontend

A193174 Duplicate of A126084.

0, 2, 1, 4, 3, 8, 5, 20, 7, 16, 13, 18, 55, 30, 53, 26, 47, 20, 41

Offset: 0

dead

A199398 XOR of the first n odd numbers.

1, 2, 7, 0, 9, 2, 15, 0, 17, 2, 23, 0, 25, 2, 31, 0, 33, 2, 39, 0, 41, 2, 47, 0, 49, 2, 55, 0, 57, 2, 63, 0, 65, 2, 71, 0, 73, 2, 79, 0, 81, 2, 87, 0, 89, 2, 95, 0, 97, 2, 103, 0, 105, 2, 111, 0, 113, 2, 119, 0, 121, 2, 127, 0, 129, 2, 135, 0, 137, 2, 143, 0, 145, 2, 151, 0, 153, 2, 159, 0, 161, 2, 167, 0, 169, 2, 175, 0, 177, 2, 183, 0, 185, 2, 191

Offset: 1

Paul D. Hanna, Nov 05 2011

			a(2) = 1 XOR 3 = 2; a(3) = 1 XOR 3 XOR 5 = 7; a(4) = 1 XOR 3 XOR 5 XOR 7 = 0.

Indranil Ghosh, Table of n, a(n) for n = 1..25000

Cf. A126084 (XOR of first n primes).

a := proc(n) local u,b,w,k;
u := 1; w := 1; b := true;
for k from 2 to n do
   u := u + 2;
   w := u + `if`(b, -w, +w);
   b := not b;
od; w end:
seq(a(n), n=1..95); # Peter Luschny, Dec 31 2014

With[{c=Range[1,201,2]},Table[BitXor@@Take[c,n],{n,100}]] (* Harvey P. Dale, Nov 19 2011 *)

PARI
```
a(n)=if(n==1,1,bitxor(a(n-1),2*n-1))
```

Vec((1 + 2*x + 6*x^2 - 2*x^3 + x^4)/(1-x^2)/(1-x^4)+O(x^99)) \\ Charles R Greathouse IV, Dec 31 2014

from operator import xor
from functools import reduce
def A199398(n): return reduce(xor,range(1,n<<1,2)) # Chai Wah Wu, Jul 09 2022

G.f.: x*(1 + 2*x + 6*x^2 - 2*x^3 + x^4)/((1-x^2)*(1-x^4)).

A293927 Numbers n such that prime(k) XOR prime(k+1) XOR ... XOR prime(n) = 0 for some k < n (where XOR denotes the binary XOR operator, and prime(n) = A000040(n)).

Original entry on oeis.org

17, 28, 30, 33, 36, 43, 45, 47, 51, 52, 56, 58, 65, 66, 72, 74, 76, 80, 84, 90, 94, 107, 111, 119, 126, 129, 130, 133, 137, 143, 145, 155, 156, 166, 169, 174, 179, 185, 192, 200, 202, 204, 208, 213, 214, 216, 219, 228, 238, 246, 248, 249, 250, 254, 258, 262

Offset: 1

Rémy Sigrist, Oct 21 2017

Equivalently, numbers n such that A126084(n) = A126084(m) for some m < n.

See A293983(n) for the least k such that prime(k) XOR prime(k+1) XOR ... XOR prime(a(n)) = 0.

			prime(33) XOR prime(34) XOR prime(35) XOR prime(36) = 137 XOR 139 XOR 149 XOR 151 = 0, hence 36 appears in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A126084, A293983.

N:= 1000: # to get all terms <= N
R[0]:= 0: T:= 2: p:= 2;
Res:= NULL:
for n from 2 to N do
  p:= nextprime(p);
  T:= Bits:-Xor(T,p);
  if assigned(R[T]) then Res:= Res, n
  else R[T]:= n
  fi
od:
Res; # Robert Israel, Oct 22 2017

s = 0; seen = 2^0; for (i = 1, 262, s = bitxor(s, prime(i)); if (bittest(seen, s), print1 (i ", "), seen += 2^s))

A293983 a(n) = least k > 0 such that prime(k) XOR prime(k+1) XOR ... XOR prime(A293927(n)) = 0 (where XOR denotes the binary XOR operator, and prime(n) = A000040(n)).

Original entry on oeis.org

8, 19, 15, 26, 33, 30, 26, 38, 22, 49, 47, 45, 58, 63, 69, 63, 65, 65, 71, 69, 69, 92, 92, 88, 123, 86, 123, 80, 132, 140, 80, 70, 153, 161, 56, 155, 176, 182, 145, 195, 143, 185, 133, 202, 125, 123, 216, 225, 235, 121, 237, 246, 235, 219, 227, 105, 260, 254

Offset: 1

Rémy Sigrist, Oct 21 2017

If a(n) = 1, then A126084(A293927(n)) = 0.
For any n > 0, A293927(n) - a(n) + 1 >= 4 (we need to XOR at least 4 consecutive prime numbers in order to obtain 0).
For any n > 0, if a(n) > 1 then A293927(n) - a(n) + 1 is even (we need to XOR an even number of odd prime numbers in order to obtain 0).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000040, A126084, A293927.

prev = vector(1774); s = 0; pi = 0; n = 0; forprime (p=1, 1697, pi++; s = bitxor(s, p); if (s==0 || prev[s], n++; print1 (prev[s]+1 ", "), prev[s] = pi));

cp's OEIS Frontend

A193174 Duplicate of A126084.

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A199398 XOR of the first n odd numbers.

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A293927 Numbers n such that prime(k) XOR prime(k+1) XOR ... XOR prime(n) = 0 for some k < n (where XOR denotes the binary XOR operator, and prime(n) = A000040(n)).

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A293983 a(n) = least k > 0 such that prime(k) XOR prime(k+1) XOR ... XOR prime(A293927(n)) = 0 (where XOR denotes the binary XOR operator, and prime(n) = A000040(n)).

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