A300168 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.

65, 2210, 58565, 4678570, 442765765, 5279766245, 2419804247185, 271780381692170, 28579081466859170, 4069607103295265285

Offset: 1

Author

Hugo Pfoertner, Feb 27 2018

Keywords

• nonn
• more
• hard

Comments

All ten known terms are squarefree. - Ray Chandler, Dec 31 2019

a(11) <= 1035219700200622531985 which is squarefree and expressible in 2047 ways. - Ray Chandler, Dec 24 2019

a(12) <= 4431331071224333359263505 which is squarefree and expressible in 4095 ways. - Ray Chandler, Dec 25 2019

Examples

			a(1) = 65 = 8^2 + 1 is the smallest expressible number (65 = 7^2 + 4^2),
a(2) = 2210 is expressible in 3 ways (2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2),
a(3) = 58565 is expressible in 7 ways,
a(4) = 4678570 is expressible in 15 ways,
a(5) = 442765765 is expressible in 31 ways.
Would a(6) be expressible in 63 ways?
Yes, a(6) = 5279766245 is expressible in 63 ways. - _Hugo Pfoertner_, Mar 02 2018
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
a(8) = 271780381692170 can be expressed in 255 ways.
a(9) = 28579081466859170 can be expressed in 511 ways.
a(10) = 4069607103295265285 can be expressed in 1023 ways.
		

Crossrefs

Cf. A300162, A300166, A300167.

Extensions

a(7) from Robert Price, Mar 11 2018

a(7) corrected, a(8)-a(9) added by Ray Chandler, Dec 23 2019

a(10) added by Ray Chandler, Dec 31 2019