A297961 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 alternately left- and right-concatenating a digit to the a(n-1) primes obtained in the previous iteration.

4, 11, 20, 53, 51, 100, 63, 76, 42, 43, 20, 13, 4, 4, 1

Offset: 1

Author

Seiichi Manyama, Jan 09 2018

Keywords

• nonn
• full
• base
• fini

Comments

No 16-digit numbers can be obtained from the 15-digit number 889292677731979.

Examples

			1-digit   2-digit    3-digit    4-digit   ...  15-digit
---------------------------------------------------------------
2
3         13         131        2131
                                6131
                     137        2137
                                3137
                                9137
                     139        4139
          23         233        5233
                                8233
                     239        2239
                                9239
          43         431        5431
                                8431
                                9431
                     433        1433
                                3433
                                7433
                                9433
                     439        1439
                                9439
          53
          73         733        1733
                                3733
                                4733
                                6733
                                9733
                     739        3739
                                9739
          83         839        5839
                                8839
                                9839
5
7         17         173        6173
                                9173
                     179        2179
                                5179
                                8179
          37         373        1373
                                3373
                                4373
                                6373
                     379        6379
          47         479        5479
                                9479
          67         673        3673
                                4673
                                6673
                                7673
                     677        2677            889292677731979
                                3677
                                8677
                                9677
          97         971        2971
                                6971
                                8971
                     977        6977
---------------------------------------------------------------
a(1) = 4, a(2) = 11, a(3) = 20, a(4) = 53, ..., a(15)= 1.
		

Links

• Prime Curios, 889292677731979

Crossrefs

Cf. A050986, A050987, A297960, A298048.

Programs

Mathematica

Block[{b = 10, t}, t = Select[Range[b], CoprimeQ[#, b] &]; TakeWhile[Length /@ Fold[Function[{a, n}, Append[a, If[EvenQ[n], Join @@ Map[Function[k, Select[Map[Prepend[k, #] &, Range[9]], 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, 16]], # > 0 &]] (* Michael De Vlieger, Jan 20 2018 *)

Python

from sympy import isprime
def alst():
  primes, alst = [2, 3, 5, 7], []
  while len(primes) > 0:
    alst.append(len(primes))
    if len(alst)%2 == 1:
      candidates = set(int(d+str(p)) for p in primes for d in "123456789")
    else:
      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