A295520 -id:A295520 - cp's OEIS frontend

A295793 a(n) is the least k such that A295520(k) = n.

2, 4, 0, 8, 25, 24, 35, 34, 201, 200, 203, 202, 297, 296, 299, 298, 1335, 1334, 1333, 1332, 1331, 1330, 1329, 1328, 3295, 3294, 3293, 3292, 3291, 3290, 3289, 3288, 11749, 11748, 11761, 11760, 11745, 11744, 11765, 11764, 11757, 11756, 19623, 19622, 11753, 11752, 19619, 19618, 25475, 25474, 25473, 25472

Offset: 0

Robert Israel, Nov 27 2017

			a(3)=8 because A295520(8)=3 and this is the first appearance of 3 in A295520.

Robert Israel, Table of n, a(n) for n = 0..175

Cf. A295520.

N:= 100: # to get a(0)..a(N)
A295520:= proc(n) local k;
  for k from 0 do if isprime(Bits:-Xor(k,n)) then return k fi od
end proc:
V:= Array(0..N,-1):
count:= 0:
for n from 0 while count < N+1 do
r:= A295520(n);
if r <= N and V[r]=-1 then
   count:= count+1; V[r]:= n
fi
od:
convert(V,list); # Robert Israel, Nov 27 2017

With[{s = Array[Block[{k = 0}, While[! PrimeQ@ BitXor[k, #], k++]; k] &, 10^6]}, FirstPosition[s, #][[1]] /. 1 -> 0 & /@ Take[#, LengthWhile[Differences@ #, # == 1 &]] &@ Union@ s] (* Michael De Vlieger, Nov 27 2017 *)

from itertools import count
from sympy import isprime
def A295793(n): return next(k for k in count(0) if next((m for m in range(n+1) if isprime(k^m)),None)==n) # Chai Wah Wu, Aug 23 2023

A330270 a(n) is the least nonnegative integer k such that n XOR k is a square (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

0, 0, 2, 2, 0, 1, 2, 3, 1, 0, 3, 2, 5, 4, 7, 6, 0, 1, 2, 3, 4, 5, 6, 7, 1, 0, 3, 2, 5, 4, 7, 6, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1

Offset: 0

Rémy Sigrist, Dec 08 2019

This sequence has similarities with A329794 as the XOR operator and the "box" operator defined in A329794 both map (n, n) to 0 for any n (however here we accept 0 as a square).

			For n = 7,
- 7 XOR 0 = 7 (not a square),
- 7 XOR 1 = 6 (not a square),
- 7 XOR 2 = 5 (not a square),
- 7 XOR 3 = 4 = 2^2,
- hence a(7) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the ordinal transform of the first 2^20 terms
Rémy Sigrist, Scatterplot of (x, y) such that x XOR y is a square, 0 <= x, y <= 1023

See A330271 for the cube variant.

Cf. A000290, A003987, A295520, A329794.

A330270[n_] := Module[{k = -1}, While[!IntegerQ[Sqrt[BitXor[n, ++k]]]]; k];
Array[A330270, 100, 0] (* Paolo Xausa, Feb 19 2024 *)

a(n) = for (k=0, oo, if (issquare(bitxor(n, k)), return (k)))

from itertools import count
from sympy.ntheory.primetest import is_square
def A330270(n): return next(k for k in count(0) if is_square(n^k)) # Chai Wah Wu, Aug 22 2023

a(n) = 0 iff n is a square.

a(a(n)) <= n.

A295335 a(n) = least k >= 0 such that n OR k is prime (where OR denotes the bitwise OR operator).

Original entry on oeis.org

2, 2, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 17, 16, 1, 0, 1, 0, 3, 2, 1, 0, 5, 4, 5, 4, 1, 0, 1, 0, 5, 4, 9, 8, 1, 0, 9, 8, 1, 0, 1, 0, 3, 2, 1, 0, 5, 4, 9, 8, 1, 0, 73, 72, 3, 2, 1, 0, 1, 0, 65, 64, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 5, 4, 3, 2, 1, 0, 3, 2, 1, 0, 43

Offset: 0

Rémy Sigrist, Nov 23 2017

See A295609 for the corresponding prime numbers.
We can show that this sequence is well defined by using Dirichlet's theorem on arithmetic progressions.
a(n) = 0 iff n is prime.
For any n >= 0, n AND a(n) = 0 (where AND denotes the bitwise AND operator).
If a(n) = x + y with x AND y = 0, then a(n + x) = y.
This sequence has similarities with A007920: here we check n OR k, there we check n + k.
See A295520 for the XOR variant.
For any k > 0, a(2^(6*k)-1) >= 2^(6*k) (hence the sequence is unbounded).

			For n = 42, 42 OR 0 = 42 is not prime, 42 OR 1 = 43 is prime, hence a(42) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Rémy Sigrist, Scatterplot of the first 2^17 terms
Index entries for sequences related to binary expansion of n

Cf. A003986, A007920, A295520, A295609.

Table[Block[{k = 0}, While[! PrimeQ@ BitOr[k, n], k++]; k], {n, 0, 84}] (* 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 (k)))

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

A344220 a(n) is the least k >= 0 such that n XOR k is a binary palindrome (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, May 12 2021

The binary expansions of n and of n XOR a(n) have the same length, say k, and their first ceiling(k/2) bits are the same.

			For n=42:
- 42 XOR 0 = 42 ("101010" in binary) is not a binary palindrome,
- 42 XOR 1 = 43 ("101011" in binary) is not a binary palindrome,
- 42 XOR 2 = 40 ("101000" in binary) is not a binary palindrome,
- 42 XOR 3 = 41 ("101001" in binary) is not a binary palindrome,
- 42 XOR 4 = 46 ("101110" in binary) is not a binary palindrome,
- 42 XOR 5 = 47 ("101111" in binary) is not a binary palindrome,
- 42 XOR 6 = 44 ("101100" in binary) is not a binary palindrome,
- 42 XOR 7 = 45 ("101101" in binary) is a binary palindrome,
- so a(42) = 7.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Scatterplot of the ordinal transform of the first 2^18 terms
Rémy Sigrist, Scatterplot of (x, y) such that x XOR y is a binary palindrome and x, y < 1024

Cf. A006995, A070939, A295520, A330270, A344259.

A344220[n_] := Module[{k = -1}, While[!PalindromeQ[IntegerDigits[BitXor[n, ++k], 2]]];k]; Array[A344220, 100, 0] (* Paolo Xausa, Feb 19 2024 *)

a(n) = my (b); for (k=0, oo, if ((b=binary(bitxor(n, k)))==Vecrev(b), return (k)))

from itertools import count
def A344220(n): return next(k for k in count(0) if (s := bin(n^k)[2:])[:(t:=len(s)+1>>1)]==s[:-t-1:-1]) # Chai Wah Wu, Aug 23 2023

a(n) = 0 iff n belongs to A006995.
A070939(n XOR a(n)) = A070939(n).
A344259(n XOR a(n)) = A344259(n).

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A295793 a(n) is the least k such that A295520(k) = n.

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A330270 a(n) is the least nonnegative integer k such that n XOR k is a square (where XOR denotes the bitwise XOR operator).

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A295335 a(n) = least k >= 0 such that n OR k is prime (where OR denotes the bitwise OR operator).

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Mathematica

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A344220 a(n) is the least k >= 0 such that n XOR k is a binary palindrome (where XOR denotes the bitwise XOR operator).

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Comments

Examples

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Crossrefs

Programs

Mathematica

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Python

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