A363761 -id:A363761 - cp's OEIS frontend

A363762 Numbers k for which A363763(k) = -1.

46, 55, 62, 71, 80, 86, 107, 130, 172, 187, 195, 208, 222, 247, 259, 263, 268, 272, 280, 297, 314, 330, 358, 363, 370, 372, 379, 394, 400, 405, 429, 449, 450, 462, 489, 500, 529, 534, 587, 629, 641, 646, 652, 667, 668, 672, 704, 715, 733, 736, 749, 769, 775, 776, 778, 785, 793, 799

Offset: 1

Hugo Pfoertner, Jun 20 2023

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..2400

Cf. A001481, A004018, A363522, A363761.

Numbers not occurring as terms of A077773.

from itertools import count, islice
from sympy import factorint
def A363762_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        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:
                break
        else:
            yield n
A363762_list = list(islice(A363762_gen(),20)) # Chai Wah Wu, Jun 22 2023

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.

Original entry on oeis.org

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

Hugo Pfoertner, Jun 20 2023

sign

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

			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)

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

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.

\\ 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)

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

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

A363522 Number of integers k such that there are exactly n distinct numbers j with k^2 < j < (k+1)^2 which can be expressed as sum of two squares.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 3, 2, 1, 1, 3, 2, 2, 1, 3, 3, 1, 3, 1, 2, 1, 4, 2, 1, 3, 1, 3, 1, 3, 1, 4, 1, 3, 2, 1, 1, 4, 1, 4, 2, 3, 0, 2, 3, 3, 3, 2, 2, 2, 1, 0, 3, 5, 1, 4, 1, 4, 0, 2, 2, 3, 4, 1, 1, 3, 3, 0, 5, 1, 4, 1, 2, 1, 3, 4, 0, 3, 3, 2, 2, 4, 0, 3

Offset: 0

Rainer Rosenthal, Jul 07 2023

nonn

Number of occurrences of n in A077773.

			a(0) = 1, since A077773(k) = 0 at the single index k = 0.
a(6) = 3, since A077773(k) = 6 for these 3 indices: k = 8, 9, and 11.
a(46) = 0, since A077773 doesn't contain 46; see A363761, A363762 and A363763.

Rainer Rosenthal, Table of n, a(n) for n = 0..10000

Cf. A077773, A363761, A363762, A363763.

from sympy import factorint
def A363522(n):
    s = 0
    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:
            s += 1
    return s # Chai Wah Wu, Jul 10 2023

A364341 a(n) is the greatest 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.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 11, 10, 14, 12, 16, 20, 22, 23, 21, 27, 29, 30, 32, 35, 38, 37, 42, 44, 47, 43, 54, 52, 51, 58, 57, 62, 56, 71, 64, 67, 68, 73, 76, 77, 78, 83, 72, 87, 90, 91, -1, 95, 103, 100, 107, 109, 105, 104, 108, -1, 116, 119, 110, 129, 117, 126, -1, 128, 134

Offset: 0

Rainer Rosenthal, Jul 20 2023

sign

Index of last occurrence of n in A077773 if there is any, otherwise -1.

Cf. A077773, A363761, A363763.

cp's OEIS Frontend

A363762 Numbers k for which A363763(k) = -1.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A363522 Number of integers k such that there are exactly n distinct numbers j with k^2 < j < (k+1)^2 which can be expressed as sum of two squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A364341 a(n) is the greatest 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.

Views

Author

Keywords

Comments

Crossrefs