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

1,1

Comments

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

Links

Table of n, a(n) for n=1..31.

Example

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.

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 *)

Prog

(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)
(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

Crossrefs

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

Sequence in context: A231358 A000303 A351186 * A344267 A144316 A180145

Adjacent sequences: A298045 A298046 A298047 * A298049 A298050 A298051

Keyword

nonn,fini,full,base

Author

Seiichi Manyama, Jan 11 2018

Extensions

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

Status

approved