|
%I #25 Jul 06 2024 13:52:17
%S 1,3,8,11,12,14,17,23,30,33,35,37,41,48,59,60,65,68,77,79,82,84,89,93,
%T 94,99
%N Numbers k with property that there exists an m>k such that the sum of the natural numbers from k^2 to m^2 inclusive is a square number.
%H Nicolay Avilov, <a href="https://www.diofant.ru/problem/3639/">Problem 1876. Segment of a natural series</a> (in Russian).
%e The number 3 is a member of the sequence because the sum of all natural numbers from 3^2 to 4^2 inclusive is 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100, with 100 = 10^2.
%o (PARI) check(k, mm=100) = my(d=2*k^2-1, v=List([]), x, y, z); for(t=d+1, 17*d, if(issquare((t^2-d^2)/2), listput(v, t))); if(v[#v\2] != 3*d, return(-1)); for(i=1, #v\2, x=v[i]; y=v[i+#v\2]; for(j=1, mm, if(issquare((x-1)/2) && x>d+2, return(1)); z=6*y-x; x=y; y=z)); 0; \\ _Jinyuan Wang_, Jul 06 2024; just for checking
%Y Cf. A000290, A372631, A373330.
%K nonn,more
%O 1,2
%A _Nicolay Avilov_, May 07 2024
|
