A363761 a(n) is the least k < 3*n such that there are exactly n distinct numbers j that can be expressed as sum of two squares with k^2 < j < (k+1)^2, or -1 if such a k does not exist.

0, 1, 2, 4, 5, 7, 8, 10, 13, 12, 15, 17, 19, 23, 21, 24, 25, 28, 32, 31, 34, 37, 39, 44, 41, 43, 45, 50, 51, 48, 57, 55, 56, 59, 64, 63, 68, 69, 74, 77, 78, 75, 72, 80, 88, 84, -1, 94, 89, 96, 93, 99, 97, 102, 108, -1, 106, 111, 110, 113, 117, 120, -1, 123, 133, 127, 130, 137, 142, 138, 139, -1, 135

Offset: 0

Author

Hugo Pfoertner, Jun 22 2023

Keywords

• sign

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A001481, A004018, A077773, A363762.

Identical with A363763 for n <= 11459, but increasingly different afterwards, i.e., a(11460) = -1, whereas A363763(11460) = 34451.

Programs

PARI

a363761(upto) = {for (n=0, upto, my(kfound=-1);
for (k=0, 3*n, my(k1=k^2+1, k2=k*(k+2), m=0);
for (j=k1, k2, m+= (sumdiv(j,d, (d%4==1)-(d%4==3))>0); if (m>n, break));
if (m==n, kfound=k; break); if (m==n, kfound=k; break)); print1(kfound,", "))};
a363761(75)

Python

from sympy import factorint
def A363761(n):
    for k in range(n>>1,3*n):
        c = 0
        for m in range(k**2+1,(k+1)**2):
            if all(p==2 or p&3==1 or e&1^1 for p, e in factorint(m).items()):
                c += 1
                if c>n:
                    break
        if c==n:
            return k
    return -1 # Chai Wah Wu, Jun 22 2023

Formula

If a(n) != -1, then a(n) >= n/2. - Chai Wah Wu, Jun 22 2023

a(n) = A363763(n) for n <= 11459.

a(n) = -1 for n > 15898.