A358421 Primes that are the concatenation of two primes with the same number of digits.

23, 37, 53, 73, 1117, 1123, 1129, 1153, 1171, 1319, 1361, 1367, 1373, 1723, 1741, 1747, 1753, 1759, 1783, 1789, 1913, 1931, 1973, 1979, 1997, 2311, 2341, 2347, 2371, 2383, 2389, 2917, 2953, 2971, 3119, 3137, 3167, 3719, 3761, 3767, 3779, 3797, 4111, 4129, 4153, 4159, 4337, 4373, 4397, 4723, 4729

Offset

1,1

Comments

It appears that there are ~ 0.81*100^k/(k^2 log^2 10) 2k-digit numbers in this sequence, making their relative density 0.9/(k^2 log^2 10) among 2k-digit numbers. Of course there are no 2k+1-digit terms in the sequence. - Charles R Greathouse IV, Nov 15 2022

The second Mathematica program below generates all 2-digit and 4-digit terms of the sequence. To generate all 2,753 6-digit terms of the sequence, use this Mathematica program: Select[1000#[[1]]+#[[2]]&/@Tuples[Prime[Range[26,168]],2],PrimeQ]. There are 112,649 8-digit terms of the sequence. - Harvey P. Dale, Feb 28 2023

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(5) = 1117 is a term because 11 and 17 are both 2-digit primes and 1117 is prime.

Maple

Res:= NULL: count:= 0:
for d from 2 by 2 while count < 100 do
  pq:= 10^(d-1);
  while count < 100 do
    pq:= nextprime(pq);
    if pq > 10^d then break fi;
    q:= pq mod 10^(d/2);
    if q < 10^(d/2-1) then next fi;
    p:= (pq-q)/10^(d/2);
    if isprime(p) and isprime(q) then Res:= Res, pq; count:= count+1 fi
od od:
Res;

Mathematica

Select[Prime[Range[640]], EvenQ[(s = IntegerLength[#])] && IntegerDigits[#][[s/2 + 1]] > 0 && And @@ PrimeQ[QuotientRemainder[#, 10^(s/2)]] &] (* Amiram Eldar, Nov 15 2022 *)
Join[Select[FromDigits/@Tuples[Prime[Range[4]], 2], PrimeQ], Select[100#[[1]]+ #[[2]]&/@ Tuples[Prime[Range[5, 25]], 2], PrimeQ]] (* Harvey P. Dale, Feb 28 2023 *)

Prog

(Python)
from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    for d in count(1):
        pow = 10**d
        for p in primerange(10**(d-1), pow):
            for q in primerange(10**(d-1), pow):
                t = p*pow + q
                if isprime(t): yield t
print(list(islice(agen(), 51))) # Michael S. Branicky, Nov 15 2022

Crossrefs

Subsequence of A000040 and of A001637.

Sequence in context: A124888 A141521 A080906 * A237766 A215163 A089685

Adjacent sequences: A358418 A358419 A358420 * A358422 A358423 A358424

Keyword

nonn,base,easy

Author

J. M. Bergot and Robert Israel, Nov 15 2022

Status

approved