

A295609


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


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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: A151570 A059036 A184442 * A163466 A306733 A085207
Adjacent sequences: A295606 A295607 A295608 * A295610 A295611 A295612


KEYWORD

nonn,base


AUTHOR

Rémy Sigrist, Nov 24 2017


STATUS

approved



