A354367 Successive pairs of terms (a, b) such that (a + b) is a square and at least one of a and b is a prime number. This is the lexicographically earliest infinite sequence of distinct terms > 0 with this property.

1, 3, 2, 7, 4, 5, 6, 19, 8, 17, 10, 71, 11, 14, 12, 13, 15, 181, 18, 31, 20, 29, 21, 43, 22, 59, 23, 26, 24, 97, 27, 37, 28, 53, 30, 139, 32, 89, 33, 67, 34, 47, 35, 109, 38, 83, 39, 61, 40, 41, 42, 79, 44, 317, 45, 151, 46, 179, 48, 73, 50, 239, 51, 349, 52, 173, 54, 307, 55, 269, 56, 113, 57, 199, 58

Offset: 1

Author

Eric Angelini and Carole Dubois, May 24 2022

Keywords

• nonn

Comments

This is not a permutation of the integers > 0 as no square > 4 will appear. Two prime terms can form a pair (2 and 7 for instance) but at least one term must be prime [the pair (1, 3) is ok].

Examples

			The earliest pairs with their square sum: (1, 3) = 4, (2, 7) = 9, (4, 5) = 9, (6, 19) = 25, (8, 17) = 25, (10, 71) = 81, (11, 14) = 25, (12, 13) = 25, etc.
		

Links

• Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..10^4, showing records in red, numbers entering late in blue, highlighting primes in green, fixed points in gold, and composite prime powers in magenta.

Crossrefs

Cf. A354368, A354369, A354370 (same idea).

Programs

Mathematica

nn = 10^4; c[] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; If[EvenQ[i], While[Nand[c[k] == 0, AnyTrue[{#, k}, PrimeQ], IntegerQ@ Sqrt[# + k]] &[a[i - 1]], k++]]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[Or[c[u] > 0, And[IntegerQ@ Sqrt@ u, u > 4]], u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 24 2022 *)