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

0,1

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000040, A030996, A030997.

Sequence in context: A300832 A329350 A283450 * A127012 A125503 A127009

Adjacent sequences: A297932 A297933 A297934 * A297936 A297937 A297938

Keyword

nonn,base,easy

Author

Paolo P. Lava, Jan 09 2018

Status

approved