A328151 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + y^2 = k.

0, 2, 5, 50, 65, 1250, 325, 31250, 1105, 8450, 8125, 19531250, 5525, 488281250, 105625, 211250, 27625, 305175781250, 71825, 7629394531250, 138125, 5281250, 126953125, 4768371582031250, 160225, 35701250, 1221025, 2442050, 3453125

Offset: 0

Author

Felix Fröhlich, Oct 05 2019

Keywords

• nonn
• hard
• more

Comments

a(n) is the smallest nonnegative i such that A063725(i) = n.

If a(n) exists, then a(n) is of the form 2*m^2 if and only if n is odd. - Chai Wah Wu, Jun 28 2024

Examples

			For n = 3: The sums of the two members of each of the pairs (1, 49), (25, 25) and (49, 1) is 50 and 50 is the smallest nonnegative integer where exactly 3 such pairs exist, so a(3) = 50.
		

Links

• Chai Wah Wu, terms a(n) for n odd and a(n) <= 3433037534088061250.

Crossrefs

Cf. A063725, A093195.

Programs

PARI

a063725(n) = if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ after Charles R Greathouse IV in A063725
a(n) = for(x=0, oo, if(a063725(x)==n, return(x)))

Python

# uses Python code from A063725
from itertools import count
def A328151(n): return next(m for m in ((k**2<<1) if n&1 else k for k in count(0)) if A063725(m)==n) # Chai Wah Wu, Jun 28 2024

Formula

Conjecture: a(2k) = A093195(k) for k >= 1, a(2k+1) = 2*A006339(k)^2 for k >= 0. - Jon E. Schoenfield, Jan 23 2022

Extensions

a(13)-a(22) from Bert Dobbelaere, Oct 20 2019

a(23)-a(28) from Chai Wah Wu, Jun 28 2024