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Nonnegative integers that are the difference of two squares but not the sum of two squares.
3

%I #16 Jun 28 2022 11:37:31

%S 3,7,11,12,15,19,21,23,24,27,28,31,33,35,39,43,44,47,48,51,55,56,57,

%T 59,60,63,67,69,71,75,76,77,79,83,84,87,88,91,92,93,95,96,99,103,105,

%U 107,108,111,112,115,119,120,123,124,127,129,131,132,133,135,139,140

%N Nonnegative integers that are the difference of two squares but not the sum of two squares.

%C Intersection of A022544 (not the sum of two squares) and A042965 (differences of two squares).

%C The sequence contains all 4k + 3 and no 4k + 2 integers, and some 4k (4*A022544) and 4k+1 (A084109) integers. First differences are thus 1, 2, 3 or 4, each occurring infinitely often.

%H Jean-Christophe Hervé, <a href="/A263737/b263737.txt"> Table of n, a(n) for n = 1..5000</a>

%t rs[n_] := Reduce[n == x^2 + y^2, {x, y}, Integers]; rd[n_] := Reduce[0 <= y <= x && n == x^2 - y^2, {x, y}, Integers]; Reap[Do[If[rs[n] == False && rd[n] =!= False, Sow[n]], {n, 0, 140}]][[2, 1]] (* _Jean-François Alcover_, Oct 26 2015 *)

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o def A263737_gen(): # generator of terms

%o return filter(lambda n:n & 3 != 2 and any(p & 3 == 3 and e & 1 for p, e in factorint(n).items()),count(0))

%o A263737_list = list(islice(A263737_gen(),30)) # _Chai Wah Wu_, Jun 28 2022

%Y Cf. A001481, A022544, A042965, A016825, A020668, A062316, A097269, A263715.

%K nonn

%O 1,1

%A _Jean-Christophe Hervé_, Oct 25 2015