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A346786
If a(n) is prime, then a(n) + a(n+1) is prime; if a(n) is not prime, then a(n) + a(n+1) is not prime. This is also true for pairs of consecutive digits: if the first one is prime, the sum of the pair is also prime; if the first one is nonprime, the sum of the pair is nonprime. This is the lexicographically earliest sequence of distinct terms > 0 with this property.
0
2, 1, 3, 4, 5, 6, 8, 7, 40, 9, 13, 46, 20, 10, 15, 21, 30, 18, 17, 42, 32, 19, 34, 23, 44, 25, 29, 38, 48, 45, 60, 62, 50, 64, 52, 56, 63, 80, 66, 68, 70, 82, 58, 74, 69, 76, 84, 81, 87, 401, 86, 88, 100, 90, 93, 201, 91, 95, 200, 96, 99, 97, 406, 204, 206, 208, 101, 300
OFFSET
1,1
EXAMPLE
a(1) = 2 (prime) and a(1) + a(2) = 2 + 1 = 3 [which is prime, like a(1)];
a(2) = 1 (nonprime) and a(2) + a(3) = 1 + 3 = 4 [which is nonprime, like a(2)];
a(3) = 3 (prime) and a(3) + a(4) = 3 + 4 = 7 [which is prime, like a(3)];
a(4) = 4 (nonprime) and a(4) + a(5) = 4 + 5 = 9 [which is nonprime, like a(4)];
a(5) = 5 (prime) and a(5) + a(6) = 5 + 6 = 11 [which is prime, like a(5)];
(...)
a(8) = 7 (prime) and a(8) + a(9) = 7 + 40 = 47 [which is prime, like a(8)];
now we have to consider also the digits of the pair (7,4); they are "7", the last digit of a(8), and "4", the first digit of a(9): as the first digit of the pair is prime (7), the sum of this 7 and the next digit (4) has to be prime too, which is the case, 4 + 7 = 11;
a(9) = 40 (nonprime) and a(9) + a(10) = 40 + 9 = 49 [which is nonprime, like a(9)];
the next pair of digits we have to consider after (7,4) is (4,0); as 4 is nonprime, so has to be the sum 4 + 0 (which is the case as 4 + 0 = 4); etc.
MATHEMATICA
t[x_, y_]:=If[PrimeQ@x, PrimeQ[x+y], !PrimeQ[x+y]]; a[1]=2; a[n_]:=a[n]=Block[{k=1}, While[MemberQ[Array[a, n-1], k]||!And@@(t@@@Partition[Flatten[IntegerDigits/@Join[Array[a, n-1], {k}]], 2, 1])||!t@@{a[n-1], k}, k++]; k]; Array[a, 68] (* Giorgos Kalogeropoulos, May 09 2022 *)
CROSSREFS
Cf. A219110.
Sequence in context: A109920 A109919 A345903 * A239469 A282840 A370655
KEYWORD
base,nonn
AUTHOR
Eric Angelini and Carole Dubois, Aug 03 2021
STATUS
approved