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Noncube integers n such that n^2 + 1 is the sum of 2 positive cubes.
1

%I #20 Feb 01 2016 12:10:18

%S 995,1191,1608,2049,3988,4584,4818,6323,12811,14642,14806,16710,27400,

%T 27672,28881,38501,39192,44892,46401,46818,53438,63455,64086,65385,

%U 72800,81711,98146,114729,126624,142345,159013,161573,169358,184644,186363,197559,200024,203432,211307

%N Noncube integers n such that n^2 + 1 is the sum of 2 positive cubes.

%C Corresponding values of n^2 + 1 are 990026, 1418482, 2585665, 4198402, 15904145, 21013057, 23213125, ...

%C Prime terms of this sequence are 6323, 38501, 159013, 161573, ...

%C Sequence focuses on the noncube values of n in order to eliminate trivial solutions of motivation equation that is n^2 + 1 = x^3 + y^3 where x and y are positive integers.

%C Union of this sequence and all positive cubes give the integers n such that n^2 + 1 are the sum of 2 positive cubes.

%H Chai Wah Wu, <a href="/A267119/b267119.txt">Table of n, a(n) for n = 1..330</a>

%e 995 is a term because 995^2 + 1 = 990026 = 51^3 + 95^3.

%e 1191 is a term because 1191^2 + 1 = 1418482 = 85^3 + 93^3.

%e 1608 is a term because 1608^2 + 1 = 2585665 = 76^3 + 129^3.

%o (PARI) T = thueinit('z^3+1);

%o is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;

%o for(n=2, 200000, if(is(n^2+1) && ispower(n, 3) == 0, print1(n, ", ")))

%Y Cf. A002522, A003325.

%K nonn

%O 1,1

%A _Altug Alkan_, Jan 10 2016