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

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

0,1

Comments

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Rémy Sigrist, Scatterplot of the first 2^17 terms

Formula

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

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

Example

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

Mathematica

Table[Block[{p = 2}, While[BitAnd[p, n] != n, p = NextPrime@ p]; p], {n, 0, 65}] (* Michael De Vlieger, Nov 26 2017 *)

Prog

(PARI) 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)))

Crossrefs

Cf. A062383, A109613, A295335.

Sequence in context: A341520 A059036 A184442 * A163466 A306733 A085207

Adjacent sequences: A295606 A295607 A295608 * A295610 A295611 A295612

Keyword

nonn,base

Author

Rémy Sigrist, Nov 24 2017

Status

approved