A297935 Least prime k such that n concatenations of n+1 consecutive primes in base 2, starting from k, generate another prime in base 10.

2, 2, 3, 2, 19, 53, 163, 53, 167, 31, 3, 37, 743, 97, 271, 17, 3, 41, 131, 691, 97, 181, 587, 523, 227, 211, 229, 3, 1697, 151, 1009, 23, 131, 151, 3137, 1621, 71, 439, 389, 521, 811, 1039, 179, 23, 311, 193, 227, 5869, 577, 6263, 31, 1901, 113, 1439, 1451, 107

Offset: 0

Author

Paolo P. Lava, Jan 09 2018

Keywords

• nonn
• base
• easy

Examples

			a(4) = 19 because the concatenation of 19, 23, 29, 31, 37 in base 2 is concat(concat(concat(concat(10011, 10111), 11101), 11111), 100101) that is the prime 41414629 in base 10 and 19 is the least prime to have this property.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000040, A030996, A030997.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,i,k,n;
for n from 1 to q do for k from 1 to q do
a:=ithprime(k); b:=convert(a,binary,decimal);
for i from 1 to n-1 do a:=nextprime(a);
c:=convert(a,binary,decimal); b:=b*10^(ilog10(c)+1)+c; od;
a:=convert(b,decimal,binary); if isprime(a) then print(ithprime(k)); break; fi; od; od; end: P(10^3);

Mathematica

Table[Prime@ SelectFirst[Range[2^12], Function[k, PrimeQ@ FromDigits[Join @@ IntegerDigits[Prime@ Range[k, k + n], 2],2]]], {n, 0, 55}] (* Michael De Vlieger, Jan 09 2018 *)

PARI

eva(n) = subst(Pol(n), x, 10)
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
concat_primes(start, num) = my(v=[], s=""); forprime(p=start, , v=concat(v, [eva(binary(p))]); if(#v==num, break)); for(k=1, #v, s=concat(s, Str(v[k]))); eval(s)
a(n) = forprime(k=1, , if(ispseudoprime(decimal(digits(concat_primes(k, n+1)), 2)), return(k))) \\ Felix Fröhlich, Jan 09 2018