A295335 -id:A295335 - cp's OEIS frontend

A295520 a(n) is the least k >= 0 such that n XOR k is prime (where XOR denotes the bitwise XOR operator).

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

Offset: 0

Rémy Sigrist, Nov 23 2017

a(n) = n iff n is prime.
For any n >= 0, a(n) <= A295335(n).
See A295335 for the OR variant.

			For n = 44:
- 44 XOR 0 = 44 is not prime,
- 44 XOR 1 = 45 is not prime,
- 44 XOR 2 = 46 is not prime,
- 44 XOR 3 = 47 is prime,
- hence a(44) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of (x, y) such that x XOR y is prime and x, y < 1024

Cf. A295335.

f:= proc(n) local k;
  for k from 0 do if isprime(Bits:-Xor(k,n)) then return k fi od
end proc:
map(f, [$0..200]); # Robert Israel, Nov 27 2017

Table[Block[{k = 0}, While[! PrimeQ@ BitXor[k, n], k++]; k], {n, 0, 104}] (* Michael De Vlieger, Nov 26 2017 *)

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

from itertools import count
from sympy import isprime
def A295520(n): return next(k for k in count(0) if isprime(n^k)) # Chai Wah Wu, Aug 23 2023

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

A295609 a(n) = least prime number p such that p AND n = n (where AND denotes the binary AND operator).

Original entry on oeis.org

2, 3, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 31, 31, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 31, 31, 29, 29, 31, 31, 37, 37, 43, 43, 37, 37, 47, 47, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 59, 59, 53, 53, 127, 127, 59, 59, 59, 59, 61, 61, 127, 127, 67, 67

Offset: 0

Rémy Sigrist, Nov 24 2017

For any n > 0: gcd(A109613(n), A062383(n)) = 1, hence, by Dirichlet's theorem on arithmetic progressions, we have a prime number, say p, of the form A109613(n) + k * A062383(n) with k > 0; this prime number satisfies p AND n = n; also a(0) = 2, hence the sequence is well defined for any n >= 0.
a(n) = n iff n is prime.
Each prime number appears 2*k times in this sequence for some k > 0.

			a(42) = 42 + A295335(42) = 42 + 1 = 43.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the first 2^17 terms

Cf. A062383, A109613, A295335.

Table[Block[{p = 2}, While[BitAnd[p, n] != n, p = NextPrime@ p]; p], {n, 0, 65}] (* 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 (n+k)))

a(n) = n + A295335(n).

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

cp's OEIS Frontend

A295520 a(n) is the least k >= 0 such that n XOR k is prime (where XOR denotes the bitwise XOR operator).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula