A000446 Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

0, 25, 325, 1105, 4225, 5525, 203125, 27625, 71825, 138125, 2640625, 160225, 17850625, 1221025, 1795625, 801125, 1650390625, 2082925, 49591064453125, 4005625, 44890625, 2158203125, 30525625, 5928325, 303460625, 53955078125

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Could start with a(0) = 3: the smallest nonnegative integer that can be written as sum of two squares in 0 ways. - M. F. Hasler, Jul 05 2024

Examples

			a(1) = 0 because 0 is the smallest integer which is uniquely a unique sum of two squares, namely 0^2 + 0^2.
a(2) = 25 from 25 = 5^2 + 0^2 = 3^2 + 4^2.
a(3) = 325 from 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(4) = 1105 from 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).
• G. Xiao, Two squares
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000448 (similar, but "in at least n ways").

Cf. A002144, A018782, A054994.

See A016032, A093195 and A124980 for other versions.

Programs

PARI

A000446(n)=if(n>1, A124980(n), 0) \\ M. F. Hasler, Jul 06 2024

Python

A000446=lambda n: A124980(n) if n>1 else 0 # M. F. Hasler, Jul 07 2024

Formula

An algorithm to compute the n-th term of this sequence for n>1: Write each of 2n and 2n-1 as products of their divisors, in decreasing order and in all possible ways. Equate each divisor in the product to (a1+1)(a2+1)...(ar+1), so that a1>=a2>=a3>=...>=ar, and solve for the ai. Evaluate A002144(1)^a1 x A002144(2)^a2 x ... x A002144(r)^ar for each set of values determined above, then the smaller of these products is the least integer to have precisely n partitions into a sum of two squares. [Ant King, Oct 07 2010]

a(n) = min(A018782(2n-1), A018782(2n)) for n > 1.

a(n) = A124980(n) for n > 1. - M. F. Hasler, Jul 07 2024

Extensions

Better description and more terms from David W. Wilson, Aug 15 1996

Definition improved by several correspondents, Nov 12 2007