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A230312
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Squares which cannot be written as the sum of a smaller nonzero square and twice a triangular number.
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2
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1, 4, 9, 25, 49, 64, 100, 144, 169, 324, 400, 729, 784, 1089, 1369, 1764, 2025, 2209, 3025, 3364, 3600, 3844, 3969, 4225, 4489, 5329, 5625, 6084, 6400, 7225, 7744, 8100, 8464, 10404, 10609, 11025, 12544, 13225, 13924, 14400, 15625, 16384, 16900
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OFFSET
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1,2
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COMMENTS
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The conjecture a(n) = A001912(n)^2 (mentioned in the formula part) is easy. In fact, any prime divisor of 4*n^2 + 1 is congruent to 1 modulo 4 and hence it can be written as a sum of two squares. Thus 4*n^2 + 1 = (2*n)^2 + 1^2 is composite if and only if it can be written as a sum of two squares in at least two ways. So the conjecture follows immediately. - Zhi-Wei Sun, Feb 23 2014
Positive squares that are the sum of two triangular numbers in exactly one way. Note that each positive square is the sum of two consecutive triangular numbers since A000217(n) + A000217(n+1) = n*(n+1)/2 + (n+1)*(n+2)/2 = (n+1)^2. - Altug Alkan, Jul 06 2016
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LINKS
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FORMULA
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Conjecture: a(n) = A001912(n)^2, that is, squares of numbers n such that 4n^2 + 1 is prime. - Alonso del Arte, Dec 20 2013
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EXAMPLE
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16 is not in the sequence because it can be expressed as 2^2 + 2 * 6.
But there is no such expression for 25 and hence it is in the sequence.
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MATHEMATICA
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A230312 = Reap[For[k = 1, k < 200, k++, n = k^2; If[Reduce[a > 0 && b > 0 && n == a^2 + b * (b + 1), {a, b}, Integers] == False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 03 2014 *)
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PROG
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(PARI) lista(nn) = for(n=1, nn, if(isprime(4*n^2+1), print1(n^2, ", "))); \\ Altug Alkan, Jul 06 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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