A281571 Smallest k such that (the base-2 number formed by concatenating k consecutive base-2 numbers starting at n) is prime, or 0 if no such k exists.

15, 1, 1, 2, 1, 26, 1, 2, 31

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 base-2 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(k-n+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 base-10 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