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

0,3

Comments

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

Links

Table of n, a(n) for n=0..72.

Hugo Pfoertner, PARI program for calculating a single term, Jul 2023.

Rainer Rosenthal, Illustrating a(5) = 7.

Rainer Rosenthal, First terms of A363763 illustrated.

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

Example

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)

Prog

(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

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.

Sequence in context: A091627 A160830 A363761 * A189303 A192817 A229968

Adjacent sequences: A363760 A363761 A363762 * A363764 A363765 A363766

Keyword

sign

Author

Hugo Pfoertner, Jun 20 2023

Status

approved