A355814 - OEIS

%I #20 Jul 19 2022 18:23:28

%S 35,55,210,240,595,360,560,504,630,720,1295,1848,1890,1386,1680,2640,

%T 2520,3024,5600,3960,2730,4680,6160,8775,9450,5850,5460,5544,9520

%N Smallest value t such that 1/s^2 + 1/t^2 = 1/p^2 + 1/q^2 has exactly n solutions (p,q) where p,q < t; or -1 if no such t exists.

%C Terms beyond a(11) = 1295 other than a(14) = 1386, if not equal to -1, are greater than 1500.

%C a(16) <= 7735.

%C Conjecture: a(n) is divisible by 35 for odd n.

%e t = 35: (s,p,q) = (5,7,7);

%e t = 55: (s,p,q) = (10,11,22),(10,22,11);

%e t = 210: (s,p,q) = (30,42,42),(95,114,133),(95,133,114);

%e t = 240: (s,p,q) = (70,84,112),(70,112,84),(108,135,144),(108,144,135);

%e t = 595: (s,p,q) = (85,91,221),(85,119,119),(85,221,91),(210,238,357),(210,357,238);

%e t = 360: (s,p,q) = (20,24,36),(20,36,24),(30,40,45),(30,45,40),(105,126,168),(105,168,126);

%e t = 560: (s,p,q) = (45,48,126),(70,80,140),(80,112,112),(45,126,48),(70,140,80),(252,315,336),(252,336,315);

%e t = 504: (s,p,q) = (42,56,63),(54,56,189),(42,63,56),(63,72,126),(63,126,72),(112,144,168),(112,168,144),(54,189,56);

%e t = 630: (s,p,q) = (35,42,63),(35,63,42),(56,63,120),(56,120,63),(90,126,126),(140,180,210),(140,210,180),(285,342,399),(285,399,342);

%e t = 720: (s,p,q) = (40,48,72),(40,72,48),(60,80,90),(60,90,80),(165,176,396),(210,252,336),(210,336,252),(165,396,176),(324,405,432),(324,432,405).

%o (PARI) b(n) = my(v=[;],r); for(p=1, n-1, for(q=1, n-1, r=1/(1/p^2+1/q^2-1/n^2); if(r==r\1 && issquare(r), v=concat(v,[p;q])))); v

%o search_up_to(Max,lim) = my(v=vector(Max,i,-1),num); for(n=1, lim, if((num=#b(n))>0 && num<=Max && v[num]==-1, v[num]=n)); v

%Y Cf. A355812, A355813.

%K nonn,hard,more

%O 1,1

%A _Jianing Song_, Jul 18 2022

%E a(12)-a(29) from _Bert Dobbelaere_, Jul 19 2022