A077773 -id:A077773 - cp's OEIS frontend

A277193 Number of integers k in range [n^2, ((n+1)^2)-1] for which 3 = the least number of squares that add up to k (A002828).

0, 1, 1, 3, 4, 4, 6, 6, 8, 9, 9, 12, 11, 14, 15, 14, 17, 18, 19, 19, 23, 20, 24, 25, 25, 26, 29, 29, 30, 32, 32, 32, 36, 36, 37, 39, 41, 40, 42, 43, 45, 45, 47, 46, 50, 49, 50, 54, 52, 55, 56, 57, 60, 60, 63, 60, 62, 65, 68, 64, 67, 70, 72, 69, 73, 74, 75, 76, 78, 78, 80, 84, 79, 85, 84, 84, 88, 89, 90, 90, 91, 94, 94, 97, 94, 99

Offset: 0

Antti Karttunen, Oct 04 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..512

Cf. A000290, A002828, A010052, A072401, A077773, A229062, A277194.

After the initial zero, one less than A277191.

(define (A277193 n) (add (lambda (i) (* (- 1 (A010052 i)) (- 1 (A229062 i)) (- 1 (A072401 i)))) (A000290 n) (+ -1 (A000290 (+ 1 n)))))
;; Implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

Sum_{i=n^2 .. ((n+1)^2)-1} (1-A010052(i))*(1-A229062(i))*(1-A072401(i)).
Other identities. For all n >= 0:
1 + A077773(n) + a(n) + A277194(n) = 2n+1.
For n >= 1, a(n) = A277191(n)-1.

A277194 Number of integers k in range [n^2, ((n+1)^2)-1] for which 4 = the least number of squares that add up to k (A002828).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 6, 5, 5, 6, 7, 6, 8, 8, 8, 8, 8, 8, 10, 9, 10, 11, 11, 10, 11, 11, 12, 11, 13, 14, 13, 13, 13, 15, 15, 15, 15, 16, 16, 15, 17, 17, 17, 17, 17, 19, 19, 18, 19, 19, 21, 20, 21, 21, 22, 21, 21, 22, 23, 22, 23, 23, 25, 23, 24, 26, 25, 26, 26, 26, 27, 26, 27, 27, 28, 29, 28, 29, 30, 29, 30, 30, 31

Offset: 0

Antti Karttunen, Oct 04 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..512

Cf. A000290, A002828, A010052, A072401, A077773, A229062, A277192, A277193.

(define (A277194 n) (add A072401 (A000290 n) (+ -1 (A000290 (+ 1 n)))))
;; Implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{i=n^2 .. ((n+1)^2)-1} A072401(i).
Other identities.
For n >= 0, 1 + A077773(n) + A277193(n) + a(n) = 2n+1.
For n >= 1, A277192(n) = A077773(n) + a(n).

A077768 Number of times that the sum of two squares is an integer between n^2 and (n+1)^2; multiple representations are counted multiply.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 7, 7, 7, 8, 10, 10, 11, 11, 12, 13, 15, 15, 14, 18, 17, 17, 19, 19, 21, 20, 21, 23, 22, 26, 25, 26, 27, 25, 29, 27, 32, 30, 28, 33, 33, 36, 34, 33, 37, 36, 39, 38, 40, 39, 38, 42, 40, 46, 43, 45, 44, 46, 48, 47, 49, 50, 48, 50, 50, 53, 55, 52, 55, 53, 60, 57

Offset: 1

T. D. Noe, Nov 20 2002

nonn

Related to the circle problem, cf. A077770. Note that 2*a(n)-A077770(n)/4 is the characteristic sequence for the Beatty sequence A001951. See A077769 for a more restrictive case. A077773 counts multiple representations only once.

			a(8)=7 because 65=64+1, 65=49+16, 68=64+4, 72=36+36, 73=64+9, 74=49+25 and 80=64+16 are between squares 64 and 81. Note that 65 occurs twice.

Cf. A001951, A077769, A077770.

maxN=100; lst={}; For[n=1, n<=maxN, n++, cnt=0; i=n; j=0; While[i>=j, j=1; While[i^2+j^2<(n+1)^2, If[i>=j&&i^2+j^2>n^2, cnt++ ]; j++ ]; i--; j-- ]; AppendTo[lst, cnt]]; lst

A077774 Number of integers between n^2 and (n+1)^2 that are the sum of two coprime squares of opposite parity; multiple representations are counted once.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 3, 4, 3, 3, 5, 4, 4, 5, 5, 5, 5, 6, 5, 5, 7, 6, 6, 6, 8, 7, 7, 8, 9, 8, 7, 8, 9, 7, 9, 10, 7, 11, 10, 9, 10, 13, 11, 8, 11, 12, 12, 11, 11, 13, 11, 13, 12, 12, 13, 13, 13, 14, 14, 13, 14, 13, 15, 13, 15, 14, 17, 15, 14, 17, 16, 16, 16, 17, 16, 18, 18, 16, 15

Offset: 1

T. D. Noe, Nov 20 2002

nonn

See A077773 for a similar, but less restrictive sequence. A077769 counts multiple representations multiply.

			a(8)=2 because 65=64+1=49+16 and 73=64+9 are between squares 49 and 64. Note that 65 is counted only once.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A077769, A077773.

maxN=100; lst={}; For[n=1, n<=maxN, n++, sqrs={}; i=n; j=0; While[i>=j, j=1; While[i^2+j^2<(n+1)^2, If[i>=j&&i^2+j^2>n^2&&GCD[i, j]==1&&OddQ[i]==EvenQ[j], AppendTo[sqrs, i^2+j^2]]; j++ ]; i--; j-- ]; AppendTo[lst, Length[Union[sqrs]]]]; lst

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.

A364444 a(n) is the number of integers k with n^2 < k < (n+1)^2 that are the sum of no more than 3 squares, counting multiple representations only once.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 10, 11, 14, 15, 16, 18, 20, 22, 23, 24, 27, 29, 30, 31, 34, 34, 36, 38, 40, 42, 44, 44, 47, 48, 49, 51, 54, 55, 57, 58, 61, 61, 62, 65, 67, 69, 69, 71, 73, 75, 76, 78, 81, 81, 83, 85, 87, 89, 89, 91, 94, 95, 97, 97, 100, 101, 103, 104, 107, 109

Offset: 0

Hugo Pfoertner, Aug 01 2023

nonn

Cf. A004215, A077773, A364445.

isA004215(n)= n\4^valuation(n, 4)%8==7 \\ after M. F. Hasler
a364444(n) = sum (k=n^2+1, n^2+2*n, !isA004215(k))

def A364444(n): return sum(1 for k in range(n**2+1,(n+1)**2) if (m:=(~k&k-1).bit_length())&1 or (k>>m)&7<7) # Chai Wah Wu, Aug 01 2023

A364445 Complement of A364444.

Original entry on oeis.org

1, 4, 6, 9, 12, 13, 17, 19, 21, 25, 26, 28, 32, 33, 35, 37, 39, 41, 43, 45, 46, 50, 52, 53, 56, 59, 60, 63, 64, 66, 68, 70, 72, 74, 77, 79, 80, 82, 84, 86, 88, 90, 92, 93, 96, 98, 99, 102, 105, 106, 108, 112, 113, 116, 118, 119, 120, 123, 124, 127, 129, 132, 133

Offset: 1

Hugo Pfoertner, Aug 01 2023

nonn

Compared to A364444, this sequence is what A363762 is to A077773.

Cf. A004215, A077773, A363762.

from itertools import count, islice
def A364445_gen(): # generator of terms
    a = 0
    for n in count(1):
        b = sum(1 for k in range(n**2+1,(n+1)**2) if (m:=(~k&k-1).bit_length())&1 or (k>>m)&7<7)
        yield from range(a+1,b)
        a = b
A364445_list = list(islice(A364445_gen(),20)) # Chai Wah Wu, Aug 01 2023

A385754 Positive numbers not occurring in A384797.

Original entry on oeis.org

1, 6, 16, 20, 25, 30, 33, 41, 48, 53, 57, 59, 62, 67, 74, 75, 78, 86, 90, 93, 98, 100, 107, 110, 113, 114, 123, 128, 130, 135, 138, 142, 145, 151, 153, 157, 159, 162, 165, 168, 178, 183, 191, 202, 204, 211, 212, 220, 223, 229, 232, 245, 254, 255, 283, 286, 291, 301

Offset: 1

Hugo Pfoertner, Jul 08 2025

nonn

This sequence is to A384797 what A363762 is to A077773.

Cf. A047800, A077773, A363762, A384797.

cp's OEIS Frontend

A277193 Number of integers k in range [n^2, ((n+1)^2)-1] for which 3 = the least number of squares that add up to k (A002828).

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A277194 Number of integers k in range [n^2, ((n+1)^2)-1] for which 4 = the least number of squares that add up to k (A002828).

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A077768 Number of times that the sum of two squares is an integer between n^2 and (n+1)^2; multiple representations are counted multiply.

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Mathematica

A077774 Number of integers between n^2 and (n+1)^2 that are the sum of two coprime squares of opposite parity; multiple representations are counted once.

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Mathematica

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.

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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.

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A364444 a(n) is the number of integers k with n^2 < k < (n+1)^2 that are the sum of no more than 3 squares, counting multiple representations only once.

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PARI

Python

A364445 Complement of A364444.

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A385754 Positive numbers not occurring in A384797.

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