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A354367
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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.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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].
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LINKS
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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