A145827 -id:A145827 - cp's OEIS frontend

A145768 a(n) = the bitwise XOR of squares of first n natural numbers.

0, 1, 5, 12, 28, 5, 33, 16, 80, 1, 101, 28, 140, 37, 225, 0, 256, 33, 357, 12, 412, 37, 449, 976, 400, 993, 325, 924, 140, 965, 65, 896, 1920, 961, 1861, 908, 1692, 965, 1633, 912, 1488, 833, 1445, 668, 1292, 741, 2721, 512, 2816, 609, 2981, 396, 2844, 485

Offset: 0

Vladimir Reshetnikov, Oct 18 2008

Up to n=10^8, a(15) is the only zero term and a(1)=a(9) are the only terms for which a(n)=1. Can it be proved that any number can only appear a finite number of times in this sequence? [M. F. Hasler, Oct 20 2008]
Even terms occur at A014601, odd terms at A042963; A010873(a(n))=A021913(n+1). - Reinhard Zumkeller, Jun 05 2012
If squares occur, they must be at indexes != 2 or 5 (mod 8). - Roderick MacPhee, Jul 17 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
StackExchange, Perfect squares in a XOR-Sum of perfect squares

Cf. A003815, A145827, A145828, A145829, A145830, A145831. [M. F. Hasler, Oct 20 2008]
Cf. A193232.
Cf. A000290.

import Data.Bits (xor)
a145768 n = a145768_list !! n
a145768_list = scanl1 xor a000290_list  -- Reinhard Zumkeller, Jun 05 2012

A[0]:= 0:
for n from 1 to 100 do A[n]:= Bits:-Xor(A[n-1],n^2) od:
seq(A[i],i=0..100); # Robert Israel, Dec 08 2019

Rest@ FoldList[BitXor, 0, Array[#^2 &, 50]]

an=0; for( i=1,50, print1(an=bitxor(an,i^2),",")) \\ M. F. Hasler, Oct 20 2008

al(n)=local(m);vector(n,k,m=bitxor(m,k^2))

from functools import reduce
from operator import xor
def A145768(n):
    return reduce(xor, [x**2 for x in range(n+1)]) # Chai Wah Wu, Aug 08 2014

a(n)=1^2 xor 2^2 xor ... xor n^2.

A221643 Numbers k such that k^2 XOR (k+1)^2 is a square, where XOR is the bitwise XOR operator.

Original entry on oeis.org

0, 3, 4, 23, 24, 43, 44, 76, 111, 112, 115, 139, 164, 180, 183, 248, 264, 323, 327, 348, 411, 479, 480, 499, 611, 699, 747, 787, 943, 976, 1072, 1103, 1111, 1176, 1268, 1388, 1447, 1576, 1684, 1851, 1983, 1984, 2008, 2243, 2692, 3271, 3383, 3452, 3464, 3532, 3679, 3804, 3867

Offset: 1

Alex Ratushnyak, Mar 27 2013

Cf. A145827, A001737.

import math
for i in range(1<<16):
    s = (i*i) ^ ((i+1)*(i+1))
    t = int(math.sqrt(s))
    if s == t*t:
        print(str(i), end=',')

A145829 Square root of squares in A145768 (XOR of squares of the numbers 1..n).

Original entry on oeis.org

1, 4, 1, 15, 0, 16, 20, 31, 16, 49, 65, 224, 96, 96, 337, 144, 720, 400, 945, 625, 928, 828, 367, 928, 1889, 624, 2609, 3568, 3568, 2064, 10273, 1040, 545, 12384, 12639, 56800, 25812, 15119, 36, 864, 144383, 146463, 195440, 61391, 61072, 61072, 58128, 25872

Offset: 1

M. F. Hasler, Oct 20 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..63

a(n) = A000196( A145828(n)) = A000196( A145768( A145827(n))); A145828 = { a(n)^2 } = A145768 intersect A000290.

a145829 n = a145829_list !! (n-1)
a145829_list = map a000196 $ filter ((== 1) . a010052) $ tail a145768_list
-- Reinhard Zumkeller, Nov 09 2012

an = 0; Reap[ For[i = 1, i <= 10^6, i++, an = BitXor[an, i^2]; If[IntegerQ[r = Sqrt[an]], Print[r]; Sow[r]]]][[2, 1]] (* Jean-François Alcover, Oct 11 2013, translated from Pari *)

an=0; for( i=1,10^4, an=bitxor(an,i^2); issquare(an,&an) && print1(an","))

from itertools import count, islice
from sympy import integer_nthroot
def A145829gen(): # generator of terms
    m = 0
    for n in count(1):
        m ^= n**2
        a, b = integer_nthroot(m,2)
        if b: yield a
A145829_list = list(islice(A145829gen(),20)) # Chai Wah Wu, Dec 16 2021

A145828 Squares in A145768 (XOR of squares of the numbers 1...n).

Original entry on oeis.org

0, 1, 16, 1, 225, 0, 256, 400, 961, 256, 2401, 4225, 50176, 9216, 9216, 113569, 20736, 518400, 160000, 893025, 390625, 861184, 685584, 134689, 861184, 3568321, 389376, 6806881, 12730624, 12730624, 4260096, 105534529

Offset: 1

M. F. Hasler, Oct 20 2008

Chai Wah Wu, Table of n, a(n) for n = 1..97

Equals A145768 intersect A000290; a(n) = A145768(A145827(n)) = A145829(n)^2.

Reap[For[Sow[x=0]; k=1, k <= 10^4, k++, x = BitXor[x, k^2]; If[IntegerQ[ Sqrt[x]], Sow[x]]]][[2, 1]] (* Jean-François Alcover, Nov 25 2015 *)

an=0; for( i=1,10^4, an=bitxor(an,i^2); issquare(an) && print1(an","))

{a(n) = my(x, m, k); while( mMichael Somos, Aug 05 2014 */

from gmpy2 import is_square
filter(is_square, [reduce(lambda x,y:x^y, [x**2 for x in range(n)]) for n in range(1,10**4)]) # Chai Wah Wu, Aug 05 2014

Edited to start at 0 to match A145768 by Chai Wah Wu, Aug 05 2014

A145830 Indices for which A145768 (XOR of squares of the numbers 1...n) is a power of 2.

Original entry on oeis.org

1, 7, 9, 16, 47, 63

Offset: 1

M. F. Hasler, Oct 20 2008

Next term is > 10^7.
a(7) > 1.5*10^9. [Jon E. Schoenfield, Jan 14 2009]
a(7) > 10^11. [Sean A. Irvine, Aug 12 2010]

A145830 = A145768 intersect A000079. A145768(a(n)) = 2^A145831(n); See also A145827-A145829.

Position[ FoldList[ BitXor, 0, Range[10^6]^2], n_ /; IntegerQ[Log[2, n]]] - 1 // Flatten (* Jean-François Alcover, Sep 30 2013 *)

an=0; for( i=1,10^6, an=bitxor(an,i^2); an & an==1<

A145831 Log_2 ( A145768( A145830(n))).

Original entry on oeis.org

0, 4, 0, 8, 9, 8

Offset: 1

M. F. Hasler, Oct 20 2008

A145830 = A145768 intersect A000079. A145768(A145830(n)) = 2^a(n). See also A145827-A145829.

an=0; for( i=1,10^7, an=bitxor(an,i^2); an & an==1<

A220518 Numbers n such that A193232(n) is a triangular number.

Original entry on oeis.org

0, 1, 5, 9, 45, 49, 101, 153, 1569, 7163, 7171, 8162, 18974, 18976, 24467, 33490, 60290, 63046, 359539, 551494, 1769418, 2813691, 4140392, 4649729, 6675935, 9653486, 59131393, 158169499, 243345386, 588183781, 1315697727, 1387290631, 1522472645, 1879098546

Offset: 1

Alex Ratushnyak, Mar 27 2013

Numbers k such that bitwise XOR of first k triangular numbers is a triangular number.

Cf. A193232.

Cf. A145827 (bitwise XOR of first k squares is a square).

#include 
typedef unsigned long long U64;
U64 rootTriangular(U64 a) {
    U64 sr = 1L<<32, s, b;
    if (a < (sr/2)*(sr+1)) {
      sr>>=1;
      while (a < sr*(sr+1)/2)  sr>>=1;
    }
    for (b = sr>>1; b; b>>=1) {
        s = sr+b;
        if (a >= s*(s+1)/2)  sr = s;
    }
    return sr;
}
int main() {
  U64 a=0, i, t;
  for (i=0; i < 1L<<32; ++i) {
      a ^= i*(i+1)/2;
      t = rootTriangular(a);
      if (a == t*(t+1)/2)  printf("%llu\n", i);
  }
  return 0;
}

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A145768 a(n) = the bitwise XOR of squares of first n natural numbers.

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A221643 Numbers k such that k^2 XOR (k+1)^2 is a square, where XOR is the bitwise XOR operator.

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A145829 Square root of squares in A145768 (XOR of squares of the numbers 1..n).

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A145828 Squares in A145768 (XOR of squares of the numbers 1...n).

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A145830 Indices for which A145768 (XOR of squares of the numbers 1...n) is a power of 2.

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A145831 Log_2 ( A145768( A145830(n))).

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A220518 Numbers n such that A193232(n) is a triangular number.

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