A355814 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.

35, 55, 210, 240, 595, 360, 560, 504, 630, 720, 1295, 1848, 1890, 1386, 1680, 2640, 2520, 3024, 5600, 3960, 2730, 4680, 6160, 8775, 9450, 5850, 5460, 5544, 9520, 15470, 5040, 7920, 9240, 25740, 10710, 9360, 13860, 13104, 8190, 17550, 10920, 18720, 15120, 22176

Offset: 1

Author

Jianing Song, Jul 18 2022

Keywords

• nonn

Comments

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

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

Examples

			t = 35: (s,p,q) = (5,7,7);
t = 55: (s,p,q) = (10,11,22),(10,22,11);
t = 210: (s,p,q) = (30,42,42),(95,114,133),(95,133,114);
t = 240: (s,p,q) = (70,84,112),(70,112,84),(108,135,144),(108,144,135);
t = 595: (s,p,q) = (85,91,221),(85,119,119),(85,221,91),(210,238,357),(210,357,238);
t = 360: (s,p,q) = (20,24,36),(20,36,24),(30,40,45),(30,45,40),(105,126,168),(105,168,126);
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);
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);
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);
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).
		

Crossrefs

Cf. A328151, A355812, A355813.

Programs

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
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

Extensions

a(12)-a(29) from Bert Dobbelaere, Jul 19 2022

More terms from Jinyuan Wang, Jan 25 2025