A298048 a(1) = number of 1-digit primes (that is, 4: 2,3,5,7); then a(n) = number of distinct n-digit prime numbers obtained by left- or right-concatenating a digit to the a(n-1) primes obtained in the previous iteration.

4, 16, 70, 243, 638, 1450, 2819, 4951, 7629, 10677, 13267, 15182, 15923, 15796, 14369, 12547, 10291, 7939, 5703, 3911, 2559, 1595, 920, 561, 321, 167, 72, 37, 11, 6, 3

Offset: 1

Author

Seiichi Manyama, Jan 11 2018

Keywords

• nonn
• fini
• full
• base

Comments

A137812 lists the primes counted here. - Jon E. Schoenfield, Jan 28 2022

Examples

			2-digit primes:
-------------------
23   2->23 or 3->23
29   2->29
13   3->13
43   3->43
53   3->53 or 5->53
73   3->73 or 7->73
83   3->83
31   3->31
37   3->37 or 7->37
59   5->59
17   7->17
47   7->47
67   7->67
97   7->97
71   7->71
79   7->79
-------------------
a(2) = 16.
===================
3-digit primes:
[223, 233, 523, 823, 239, 229, 293, 829, 929, 113, 131, 313, 613, 137, 139, 311, 331, 431, 631, 317, 433, 443, 643, 743, 439, 353, 653, 853, 953, 173, 373, 733, 673, 773, 739, 337, 937, 379, 283, 383, 683, 883, 983, 839, 359, 593, 659, 859, 599, 617, 179, 271, 571, 971, 719, 347, 547, 647, 947, 479, 167, 367, 467, 677, 967, 197, 397, 797, 977, 997]
In the case of 223, 2->23 (adding the rightmost digit)->223 (adding the leftmost digit) and 2, 23 and 223 are prime.
In the case of 233, 2->23 (adding the rightmost digit)->233 (adding the rightmost digit) and 2, 23 and 233 are prime.
In the case of 523, 2->23 (adding the rightmost digit)->523 (adding the leftmost digit) and 2, 23 and 523 are prime.
a(3) = 70.
		

Crossrefs

Cf. A050986, A050987, A137812, A297960, A297961.

Programs

Mathematica

Block[{b = 10, t}, t = Select[Range[b], CoprimeQ[#, b] &]; TakeWhile[Length /@ Fold[Function[{a, n}, Append[a, Union[Join @@ {Join @@ Map[Function[k, Select[Map[Prepend[k, #] &, Range[b - 1]], PrimeQ@ FromDigits[#, b] &]], Last[a]], Join @@ Map[Function[k, Select[Map[Append[k, #] &, t], PrimeQ@ FromDigits[#, b] &]], Last[a]]}] ] ] @@ {#1, #2} &, {IntegerDigits[Prime@ Range@ PrimePi@ b, b]}, Range[2, 40]], # > 0 &]] (* Michael De Vlieger, Jan 21 2018 *)

Python

from sympy import isprime
def alst():
  primes, alst = [2, 3, 5, 7], []
  while len(primes) > 0:
    alst.append(len(primes))
    candidates = set(int(d+str(p)) for p in primes for d in "123456789")
    candidates |= set(int(str(p)+d) for p in primes for d in "1379")
    primes = [c for c in candidates if isprime(c)]
  return alst
print(alst()) # Michael S. Branicky, Apr 11 2021

Ruby

require 'prime'
def A298048(n)
  ary = [2, 3, 5, 7]
  a_ary = [4]
  (n - 1).times{|i|
    ary1 = []
    ary.each{|a|
      (1..9).each{|d|
        j = d * 10 ** (i + 1) + a
        ary1 << j if j.prime?
        j = a * 10 + d
        ary1 << j if j.prime?
      }
    }
    ary = ary1.uniq
    a_ary << ary.size
  }
  a_ary
end
p A298048(10)

Extensions

a(16)-a(31) from Michael De Vlieger, Jan 21 2018