A164022 a(n) = the smallest prime that, when written in binary, starts with the substring of n in binary.

2, 2, 3, 17, 5, 13, 7, 17, 19, 41, 11, 97, 13, 29, 31, 67, 17, 37, 19, 41, 43, 89, 23, 97, 101, 53, 109, 113, 29, 61, 31, 131, 67, 137, 71, 73, 37, 307, 79, 163, 41, 337, 43, 89, 181, 373, 47, 97, 197, 101, 103, 211, 53, 109, 223, 113, 229, 233, 59, 241, 61, 251, 127, 257

Offset

1,1

Comments

The argument used to prove that A018800(n) always exists applies here also. - N. J. A. Sloane, Nov 14 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Index entries for primes involving decimal expansion of n

Example

4 in binary is 100. Looking at the binary numbers that begin with 100: 100 = 4 in decimal is composite; 1000 = 8 in decimal is composite; 1001 = 9 in decimal is composite; 10000 = 16 in decimal is composite. But 10001 = 17 in decimal is prime. So a(4) = 17.

Maple

A164022 := proc(n) dgs2 := convert(n, base, 2) ; ldgs := nops(dgs2) ; for i from 1 do p := ithprime(i) ; if p >= n then pdgs := convert(p, base, 2) ; if [op(nops(pdgs)+1-ldgs.. nops(pdgs), pdgs)] = dgs2 then RETURN( p) ; fi; fi; od: end: seq(A164022(n), n=1..120) ; # R. J. Mathar, Sep 13 2009

Mathematica

With[{s = Map[IntegerDigits[#, 2] &, Prime@ Range[10^4]]}, Table[Block[{d = IntegerDigits[n, 2]}, FromDigits[#, 2] &@ SelectFirst[s, Take[#, UpTo@ Length@ d] == d &]], {n, 64}]] (* Michael De Vlieger, Sep 23 2017 *)

Crossrefs

A018800 is the base-10 analog.

Row n=1 of A262365. Cf. A108234 (number of new bits), A208241 (proper substring).

Sequence in context: A109843 A341768 A184840 * A089751 A137909 A035796

Adjacent sequences: A164019 A164020 A164021 * A164023 A164024 A164025

Keyword

base,nonn

Author

Leroy Quet, Aug 08 2009

Extensions

Corrected terms a(1) and a(2) (with help from Ray Chandler) Leroy Quet, Aug 16 2009

Extended by R. J. Mathar, Sep 13 2009

Status

approved