

A281571


Smallest k such that (the base2 number formed by concatenating k consecutive base2 numbers starting at n) is prime, or 0 if no such k exists.


2




OFFSET

1,1


COMMENTS

The first primes reached are 485398038695407, 2, 3, 37, 5, 288368629084891241583296816292460511, 7, 137, 55212283888448697916635329662406145945631873447, ...
Except for the second term, n and a(n) have the same parity, i.e., a(n) == n (mod 2). Is it proved (or can it be disproved) that the required k exists for all n? a(10), a(21), a(24), a(38), a(52), a(55) are larger than 1500, if they exist.  M. F. Hasler, Apr 26 2017
a(10) > 40000. Terms at indices 24, 38, 55, 56, 57, 60, 62, 65, 66, 76, 78, 91, 92, 95 are > 20000. A large known term is a(330) = 9376.  Hans Havermann, May 17 2017


LINKS

Table of n, a(n) for n=1..9.
Paolo P. Lava, First 100 terms (with 1 if a(n) is not presently known)


FORMULA

a(n) = 1 if n is prime.


EXAMPLE

a(1) = 15 because we have to concatenate the base2 numbers from 1 to 15 to reach the first prime. In fact concat(1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111) =
1101110010111011110001001101010111100110111101111, which is prime (in base 10 it is 485398038695407).


MAPLE

P:=proc(q) local a, b, k, n; for n from 1 to q do
if isprime(n) then print(1); else a:=convert(n, binary, decimal);
for k from n+1 to q do b:=convert(k, binary, decimal); a:=a*10^(ilog10(b)+1)+b; if isprime(convert(a, decimal, binary))
then print(kn+1); break; fi; od; fi; od; end: P(10^10);


MATHEMATICA

With[{nn = 2^10}, Table[Module[{k = n, w = IntegerDigits[n, 2]}, While[And[! PrimeQ[FromDigits[w, 2]], k  n < nn], k++; w = Join[w, IntegerDigits[k, 2]]]; If[k  n >= nn, 1, k  n + 1]], {n, 50}]] (* Michael De Vlieger, Apr 26 2017, with 1 indicating values of k > limit nn *)


PROG

(PARI) a(n, c=1, m=n)=while(!ispseudoprime(n), c++; n=n<<#binary(m++)+m); c


CROSSREFS

Cf. A000040, A047778.
Cf. A244424 for the base10 variant.
Sequence in context: A040228 A040229 A318650 * A040227 A040226 A172429
Adjacent sequences: A281568 A281569 A281570 * A281572 A281573 A281574


KEYWORD

nonn,base,more


AUTHOR

Paolo P. Lava, Jan 24 2017


EXTENSIONS

Edited by Max Alekseyev, Apr 26 2017.
Further edits from N. J. A. Sloane, Apr 26 2017
a(18) = 586, a(28) = 934, a(35) = 947, a(51) = 1325 (PRP), and further edits from M. F. Hasler, Apr 26 2017


STATUS

approved



