A326315 -id:A326315 - cp's OEIS frontend

A326316 Lexicographically earliest sequence of distinct terms such that the digits of a(n) and a(n+1) sum up to a prime and a(n) + a(n+1) is also a prime.

1, 2, 3, 4, 7, 6, 5, 8, 9, 20, 21, 22, 25, 28, 43, 24, 23, 44, 27, 26, 41, 42, 29, 60, 47, 62, 45, 64, 49, 40, 61, 46, 63, 68, 69, 80, 83, 48, 65, 66, 113, 84, 89, 110, 81, 82, 67, 112, 85, 88, 111, 86, 87, 152, 117, 116, 135, 134, 137, 114, 115, 118, 133, 130, 139, 132, 119, 138, 131, 150, 157, 136, 171, 176, 177, 170

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jun 24 2019

			The sequence starts with 1,2,3,4,7,6,5,8,9,20,21,... and we see indeed that:
the digits of {a(1); a(2)} have sum 1 + 2 = 3 (prime) and a(1) + a(2) is a prime too (3);
the digits of {a(2); a(3)} have sum 2 + 3 = 5 (prime) and a(2) + a(3) is a prime too (5);
the digits of {a(3); a(4)} have sum 3 + 4 = 7 (prime) and a(3) + a(4) is a prime too (7);
the digits of {a(4); a(5)} have sum 4 + 7 = 11 (prime) and a(4) + a(5) is a prime too (11);
the digits of {a(5); a(6)} have sum 7 + 6 = 13 (prime) and a(5) + a(6) is a prime too (13);
...
the digits of {a(9); a(10)} have sum 9 + 2 + 0 = 11 (prime) and a(9) + a(10) = 9 + 20 is a prime too (29);
the digits of {a(10); a(11)} have sum 2 + 0 + 2 + 1 = 5 (prime) and a(10) + a(11) = 20 + 21 is a prime too (41);
etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10001

Cf. A326315 (replace the word "prime" by "palindrome"), A326317 (replace the word "prime" by "square"); in A308728 only the sum of the digits is a prime.

A326317 Lexicographically earliest sequence of distinct terms such that the digits of a(n) and a(n+1) sum up to a square and a(n) + a(n+1) is also a square.

Original entry on oeis.org

1, 3, 6, 19, 150, 46, 123, 21, 15, 154, 42, 102, 67, 129, 40, 41, 103, 66, 130, 14, 22, 122, 47, 149, 20, 16, 9, 7, 2, 34, 110, 11, 25, 144, 52, 117, 79, 90, 106, 63, 133, 36, 160, 324, 205, 279, 250, 234, 295, 794, 230, 211, 113, 31, 5, 4, 32, 112, 57, 139, 30, 51, 118, 78, 91, 105, 64, 132, 12, 24, 120, 49, 147, 337, 192, 292

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jun 24 2019

			The sequence starts with 1,3,6,19,150,46,123,... and we see indeed that:
the digits of {a(1); a(2)} have sum 1 + 3 = 4 (square) and a(1) + a(2) is a square too (4);
the digits of {a(2); a(3)} have sum 3 + 6 = 9 (square) and a(2) + a(3) is a square too (9);
the digits of {a(3); a(4)} have sum 6 + 1 + 9 = 16 (square) and a(3) + a(4) = 6 + 19 is a square too (25);
the digits of {a(4); a(5)} have sum 1 + 9 + 1 + 5 + 0 = 16 (square) and a(4) + a(5) = 19 + 150 is a square too (169 --> square of 13);
the digits of {a(5); a(6)} have sum 1 + 5 + 0 + 4 + 6  = 16 (square) and a(5) + a(6) = 150 + 46 is a square too (196 --> square of 16);
the digits of {a(6); a(7)} have sum 4 + 6 + 1 + 2 + 3 = 16 (square) and a(6) + a(7) = 46 + 123 is a squaretoo (169);
etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10001

Cf. A326315 (replace the word "square" by "palindrome"), A326316 (replace the word "square" by "prime"); in A308727 only the sum of the digits is a square.

cp's OEIS Frontend

A326316 Lexicographically earliest sequence of distinct terms such that the digits of a(n) and a(n+1) sum up to a prime and a(n) + a(n+1) is also a prime.

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