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a(n) is the number of prime numbers created when concatenating all the arrangements of the decimal integers from 0 to 3*n+4.
0

%I #41 Sep 28 2020 21:47:43

%S 20,3202,2056675,3500185228

%N a(n) is the number of prime numbers created when concatenating all the arrangements of the decimal integers from 0 to 3*n+4.

%C Only 4 and every third integer after 4 can create primes when concatenating the integer arrangements of 0,...,3*n+4 as the other integer values will create numbers with digit sums divisible by 3, and hence are divisible by 3. The digit 0 is allowed to be the first digit in the number but is then ignored when determining if the remaining digits form a prime.

%e a(0) = 20 as there are twenty primes created when concatenating the integer arrangements of 0,1,2,3,4. They are 1423, 2143, 2341, 4231, 10243, 12043, 20143, 20341, 20431, 23041, 24103, 30241, 32401, 40123, 40213, 40231, 41023, 41203, 42013, 43201.

%e a(1) = 3202. The smallest prime created using integers 0..7 is 1234657 while the largest is 76540231.

%e a(2) = 2056675. The smallest prime created using integers 0..10 is 10123457689 while the largest is 987654310021.

%t Table[Count[FromDigits /@ Flatten /@ IntegerDigits /@ Permutations[Range[0, 3 n + 4]], _?PrimeQ], {n, 0, 2}] (* _Robert Price_, Sep 16 2020 *)

%t (* OR, if the above runs low on memory to store all the Permutations at once... *)

%t Table[p0 = Range[0, 3n+4]; p = NextPermutation[p0]; c = 0;

%t While[p != p0,

%t If[PrimeQ[FromDigits[Flatten[IntegerDigits /@ p]]], c++];

%t p = NextPermutation[p]]; c, {n, 0, 2}] (* _Robert Price_, Sep 16 2020 *)

%Y Cf. A000040, A295206, A006879, A050288, A088628, A185122, A176009, A099182.

%K nonn,base,more

%O 0,1

%A _Scott R. Shannon_, May 04 2020

%E a(3) from _Giovanni Resta_, May 04 2020