A363763 a(n) is the least k such that there are exactly n distinct numbers j that can be expressed as the 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 20 2023

Keywords

• sign

Comments

Index of first occurrence of n in A077773 if there is any, otherwise -1. - Rainer Rosenthal, Jul 07 2023

Examples

			From _Rainer Rosenthal_, Jul 09 2023: (Start)
a(5) = 7, since A077773(7) = 5 and A077773(n) != 5 for n < 7.
a(46) = -1, since a(46) < ((46+1)^2)/2 < 1105 and A077773(k) != 46 for all k < 1105.
See illustrations in the links section. (End)
		

Links

• Hugo Pfoertner, PARI program for calculating a single term, Jul 2023.
• Rainer Rosenthal, Illustrating a(5) = 7.
• Rainer Rosenthal, First terms of A363763 illustrated.

Crossrefs

Cf. A000415, A001481, A004018, A077773, A364341.

A363762 gives the positions of terms = -1.

Identical with A363761 up to a(11459) = 33864, but increasingly different afterwards, i.e., a(11460) = 34451, whereas A363761(11460) = -1.

Programs

PARI

\\ a4018(n) after Michael Somos
a4018(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)));
a363763 (upto) = {for (n=0, upto, my(kfound=-1); for (k=0, (n+1)^2\2+1, my(kp=k^2+1, km=(k+1)^2-1, m=0); for (j=kp, km, if (a4018(j), m++); if (m>n, break)); if (m==n, kfound=k; break)); print1 (kfound,", ");)};
a363763(75)

Python

from sympy import factorint
def A363763(n):
    for k in range(n>>1,((n+1)**2<<1)+1):
        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 20-26 2023

Formula

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

a(n) < (n+1)^2/2. - Jon E. Schoenfield and Chai Wah Wu, Jun 24-26 2023