A300168 - OEIS

%I #31 Dec 31 2019 09:27:50

%S 65,2210,58565,4678570,442765765,5279766245,2419804247185,

%T 271780381692170,28579081466859170,4069607103295265285

%N Numbers of the form n^2+1 that can be expressed as j^2+k^2, j>k>1, gcd(j,k)=1, in more ways than any smaller number of this form.

%C All ten known terms are squarefree. - _Ray Chandler_, Dec 31 2019

%C a(11) <= 1035219700200622531985 which is squarefree and expressible in 2047 ways. - _Ray Chandler_, Dec 24 2019

%C a(12) <= 4431331071224333359263505 which is squarefree and expressible in 4095 ways. - _Ray Chandler_, Dec 25 2019

%e a(1) = 65 = 8^2 + 1 is the smallest expressible number (65 = 7^2 + 4^2),

%e a(2) = 2210 is expressible in 3 ways (2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2),

%e a(3) = 58565 is expressible in 7 ways,

%e a(4) = 4678570 is expressible in 15 ways,

%e a(5) = 442765765 is expressible in 31 ways.

%e Would a(6) be expressible in 63 ways?

%e Yes, a(6) = 5279766245 is expressible in 63 ways. - _Hugo Pfoertner_, Mar 02 2018

%e a(7) = 2419804247185 can be expressed in 127 ways.  This continues the progression that a(n) can be expressed in n^2-1 ways. - _Robert Price_, Mar 11 2018, updated by _Ray Chandler_, Dec 23 2019

%e a(8) = 271780381692170 can be expressed in 255 ways.

%e a(9) = 28579081466859170 can be expressed in 511 ways.

%e a(10) = 4069607103295265285 can be expressed in 1023 ways.

%Y Cf. A300162, A300166, A300167.

%K nonn,more,hard

%O 1,1

%A _Hugo Pfoertner_, Feb 27 2018

%E a(7) from _Robert Price_, Mar 11 2018

%E a(7) corrected, a(8)-a(9) added by _Ray Chandler_, Dec 23 2019

%E a(10) added by _Ray Chandler_, Dec 31 2019