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

1,2

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

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

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

Example

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.

Prog

(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

Crossrefs

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

Cf. A002144, A018782, A054994.

See A016032, A093195 and A124980 for other versions.

Sequence in context: A020233 A020319 A000448 * A124980 A188355 A243089

Adjacent sequences: A000443 A000444 A000445 * A000447 A000448 A000449

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Definition improved by several correspondents, Nov 12 2007

Status

approved