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
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.
Programs
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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
Comments