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A230312
Squares which cannot be written as the sum of a smaller nonzero square and twice a triangular number.
2
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
OFFSET
1,2
COMMENTS
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
FORMULA
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
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(PARI) lista(nn) = for(n=1, nn, if(isprime(4*n^2+1), print1(n^2, ", "))); \\ Altug Alkan, Jul 06 2016
CROSSREFS
Cf. A001912.
Sequence in context: A063482 A277312 A069557 * A332646 A306043 A194269
KEYWORD
nonn,easy
AUTHOR
Kieren MacMillan, Dec 20 2013
STATUS
approved