A077773 Number of integers between n^2 and (n+1)^2 that are the sum of two squares; multiple representations are counted once.

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 6, 9, 8, 8, 10, 10, 11, 11, 12, 11, 14, 12, 13, 15, 16, 15, 15, 17, 16, 17, 19, 18, 19, 20, 19, 20, 21, 20, 22, 22, 24, 22, 25, 23, 26, 26, 24, 29, 26, 27, 28, 27, 29, 26, 31, 32, 30, 29, 33, 33, 31, 31, 35, 34, 35, 35, 35, 36, 37, 37, 33, 42, 37, 38

Offset: 0

Author

T. D. Noe, Nov 20 2002

Keywords

• nonn

Comments

Related to the circle problem, cf. A077770. See A077774 for a more restrictive case. A077768 counts the representations multiply.

Number of integers k in range [n^2, ((n+1)^2)-1] for which 2 = the least number of squares that add up to k (A002828). Because of this interpretation a(0)=0 was prepended to the beginning. - Antti Karttunen, Oct 04 2016

This sequence is not surjective, since, for instance, there is no n such that a(n) = 46. This follows from a bound observed by Jon E. Schoenfield, that if a(n) = m then n < ((m+1)^2)/2, and the fact that a(n) != 46 for all n < 1105. - Rainer Rosenthal, Jul 25 2023

Examples

			a(8)=6 because 65=64+1=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 is counted only once.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..10000 (terms 0..1024 from T. D. Noe and Antti Karttunen).
• Hugo Pfoertner, Table of n, a(n) for n = 0..500000
• Rainer Rosenthal, Illustrating A077773

Crossrefs

Cf. A000290, A000404, A002828, A010052, A077768, A077770, A077774, A229062, A277192, A277193, A277194.

Cf. A363762 (terms not occurring in this sequence), A363763.

Programs

Mathematica

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, AppendTo[sqrs, i^2+j^2]]; j++ ]; i--; j-- ]; AppendTo[lst, Length[Union[sqrs]]]]; lst

PARI

a(N)=s=0;for(n=N^2+1,(N+1)^2-1,f=0;r=sqrtint(n);forstep(i=r,1,-1,if(issquare(n-i*i),f=1;s=s+1;break)));s /* Ralf Stephan, Sep 17 2013 */

Python

from sympy import factorint
def A077773(n): return sum(1 for m in range(n**2+1,(n+1)**2) if all(p==2 or p&3==1 or e&1^1 for p, e in factorint(m).items())) # Chai Wah Wu, Jun 20 2023

Scheme

(define (A077773 n) (add (lambda (i) (* (- 1 (A010052 i)) (A229062 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)))))))
;; Antti Karttunen, Oct 04 2016

Formula

a(n) = Sum_{i=n^2+1..(n+1)^2-1} A229062(i). - Ralf Stephan, Sep 17 2013

From Antti Karttunen, Oct 04 2016: (Start)

For n >= 0, a(n) + A277193(n) + A277194(n) = 2n.

For n >= 1, A277192(n) = a(n) + A277194(n). (End)

Extensions

Term a(0)=0 prepended by Antti Karttunen, Oct 04 2016