A230312 - OEIS

%I #48 Dec 24 2016 09:51:50

%S 1,4,9,25,49,64,100,144,169,324,400,729,784,1089,1369,1764,2025,2209,

%T 3025,3364,3600,3844,3969,4225,4489,5329,5625,6084,6400,7225,7744,

%U 8100,8464,10404,10609,11025,12544,13225,13924,14400,15625,16384,16900

%N Squares which cannot be written as the sum of a smaller nonzero square and twice a triangular number.

%C 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

%C 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

%F 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

%e 16 is not in the sequence because it can be expressed as 2^2 + 2 * 6.

%e But there is no such expression for 25 and hence it is in the sequence.

%t 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 *)

%o (PARI) lista(nn) = for(n=1, nn, if(isprime(4*n^2+1), print1(n^2, ", "))); \\ _Altug Alkan_, Jul 06 2016

%Y Cf. A001912.

%K nonn,easy

%O 1,2

%A _Kieren MacMillan_, Dec 20 2013