A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

1, 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, 35409725, 100140625

Offset: 1

Author

Artur Jasinski, Nov 15 2006

Keywords

• nonn

Comments

The number must be strictly positive, but one of the squares may be zero, as we see from a(1) = 1 = 1^2 + 0^2 and a(2) = 25 = 3^2 + 4^2 = 5^0 + 0^2. - M. F. Hasler, Jul 07 2024

Examples

			a(3) = 325 is decomposable in 3 ways: 15^2 + 10^2 = 17^2 + 6^2 = 18^2 + 1^2.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).

Crossrefs

Cf. A002144, A018782, A054994, A118882.

See A016032, A000446 and A093195 for other versions.

Programs

PARI

A124980(n)={for(a=1, oo, A000161(a)==n && return(a))} \\ R. J. Mathar, Nov 29 2006, edited by M. F. Hasler, Jul 07 2024

PARI

PD(n, L=n, D=Vecrev(divisors(n)[^1])) = { if(n>1, concat(vector(#D, i, if(D[i] > L, [], D[i] < n, [concat(D[i], P) | P <- PD(n/D[i], D[i])], [[n]]))), [[]])}
apply( {A124980(n)=vecmin([prod(i=1, #a, A002144(i)^(a[i]-1)) | a<-concat([PD(n*2,n), PD(n*2-1)])])}, [1..44]) \\ M. F. Hasler, Jul 07 2024

Python

from sympy import divisors, isprime, prod
def PD(n, L=None): return [[]] if n==1 else [
    [d]+P for d in divisors(n)[:0:-1] if d <= (L or n) for P in PD(n//d, d)]
A2144=lambda upto=999: filter(isprime, range(5, upto, 4))
def A124980(n):
    return min(prod(a**(f-1) for a,f in zip(A2144(), P))
               for P in PD(n*2, n)+PD(n*2-1)) # M. F. Hasler, Jul 07 2024

Formula

a(n) = A000446(n), n > 1. - R. J. Mathar, Jun 15 2008

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

a(n) = min { k > 0 | A000161(k) = n }. - M. F. Hasler, Jul 07 2024

Extensions

More terms from R. J. Mathar, Nov 29 2006

Edited and extended by Ray Chandler, Jan 07 2012