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

1,2

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

Links

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

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

Example

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

Prog

(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

Crossrefs

Cf. A002144, A018782, A054994, A118882.

See A016032, A000446 and A093195 for other versions.

Sequence in context: A020319 A000448 A000446 * A188355 A243089 A079875

Adjacent sequences: A124977 A124978 A124979 * A124981 A124982 A124983

Keyword

nonn

Author

Artur Jasinski, Nov 15 2006

Extensions

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

Edited and extended by Ray Chandler, Jan 07 2012

Status

approved