A297960 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 right- and left-concatenating a digit to the a(n-1) primes obtained in the previous iteration.

4, 9, 30, 49, 99, 74, 101, 71, 72, 35, 28, 9, 4

Offset

1,1

Comments

No 14-digit numbers can be obtained from the four 13-digit numbers counted by a(13).

Links

Table of n, a(n) for n=1..13.

Example

1-digit   2-digit   3-digit    4-digit   ...  13-digit
------------------------------------------------------------
2         23        223        2237
                               2239
                    523        5231
                               5233
                               5237
                    823        8231
                               8233            6638182333331
                               8237
          29        229        2293
                               2297
                    829        8291
                               8293
                               8297
                    929        9293
3         31        131        1319
                    331        3313
                               3319
                    431
                    631        6311            5981563119937
                               6317
          37        137        1373
                    337        3371
                               3373
                    937        9371
                               9377
5         53        353        3533
                               3539
                    653
                    853        8537
                               8539
                    953        9533
                               9539
          59        359        3593
                    659        6599
                    859        8597
                               8599
7         71        271        2711
                               2713
                               2719
                    571        5711
                               5717
                    971        9719
          73        173        1733
                    373        3733
                               3739
                    673        6733            8313667333393
                               6737
                    773
          79        179
                    379        3793            2682637937713
                               3797
                    479        4793
                               4799
------------------------------------------------------------
a(1) = 4, a(2) = 9, a(3) = 30, a(4) = 49, ..., a(13) = 4.

Mathematica

Block[{b = 10, t}, t = Select[Range[b], CoprimeQ[#, b] &]; TakeWhile[Length /@ Fold[Function[{a, n}, Append[a, If[OddQ[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 *)

Prog

(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 == 0:
      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

Crossrefs

Cf. A050986, A050987, A297961, A298048.

Sequence in context: A241393 A186650 A091658 * A295910 A086688 A309295

Adjacent sequences: A297957 A297958 A297959 * A297961 A297962 A297963

Keyword

nonn,full,base,fini

Author

Seiichi Manyama, Jan 09 2018

Status

approved