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%I #50 Apr 11 2021 11:59:49
%S 4,11,20,53,51,100,63,76,42,43,20,13,4,4,1
%N 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.
%C No 16-digit numbers can be obtained from the 15-digit number 889292677731979.
%H Prime Curios, <a href="https://primes.utm.edu/curios/page.php/889292677731979.html">889292677731979</a>
%e 1-digit 2-digit 3-digit 4-digit ... 15-digit
%e ---------------------------------------------------------------
%e 2
%e 3 13 131 2131
%e 6131
%e 137 2137
%e 3137
%e 9137
%e 139 4139
%e 23 233 5233
%e 8233
%e 239 2239
%e 9239
%e 43 431 5431
%e 8431
%e 9431
%e 433 1433
%e 3433
%e 7433
%e 9433
%e 439 1439
%e 9439
%e 53
%e 73 733 1733
%e 3733
%e 4733
%e 6733
%e 9733
%e 739 3739
%e 9739
%e 83 839 5839
%e 8839
%e 9839
%e 5
%e 7 17 173 6173
%e 9173
%e 179 2179
%e 5179
%e 8179
%e 37 373 1373
%e 3373
%e 4373
%e 6373
%e 379 6379
%e 47 479 5479
%e 9479
%e 67 673 3673
%e 4673
%e 6673
%e 7673
%e 677 2677 889292677731979
%e 3677
%e 8677
%e 9677
%e 97 971 2971
%e 6971
%e 8971
%e 977 6977
%e ---------------------------------------------------------------
%e a(1) = 4, a(2) = 11, a(3) = 20, a(4) = 53, ..., a(15)= 1.
%t 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 *)
%o (Python)
%o from sympy import isprime
%o def alst():
%o primes, alst = [2, 3, 5, 7], []
%o while len(primes) > 0:
%o alst.append(len(primes))
%o if len(alst)%2 == 1:
%o candidates = set(int(d+str(p)) for p in primes for d in "123456789")
%o else:
%o candidates = set(int(str(p)+d) for p in primes for d in "1379")
%o primes = [c for c in candidates if isprime(c)]
%o return alst
%o print(alst()) # _Michael S. Branicky_, Apr 11 2021
%Y Cf. A050986, A050987, A297960, A298048.
%K nonn,full,base,fini
%O 1,1
%A _Seiichi Manyama_, Jan 09 2018