A277192 -id:A277192 - cp's OEIS frontend

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

T. D. Noe, Nov 20 2002

nonn

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

			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.

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

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

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

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

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 */

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

(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

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)

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

A277891 a(n) = number of odd numbers encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

The starting point ((n+1)^2)-1 of the iteration is included if it is odd, but the ending point (n^2)-1 is never included in the count.
a(n) = number of odd numbers on row n of A276574, after the initial zero-row.
On the average, the odd terms in A276573 (A276574) seem to occur more frequently than the even terms. (The last point in range 1..10000 where a(n) <= A277890(n) is n=862). See also comments in A277487 and the plot of ratio a(n)/A277890(n), also the plot of A277889.

			For n=6, we start iterating from k = ((6+1)^2)-1 = 48, with k -> k - A002828(k), to obtain 48 -> 45 -> 43 -> 40 -> 38 before reaching 35 (which is 6^2 - 1, an ending point and thus not included in the count), and the only odd numbers before that were 45 and 43, thus a(6) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
A. Karttunen, Ratio a(n)/A277890(n) drawn with OEIS Plot 2 tool

Cf. A000035, A000290, A002828, A010052, A077773, A260733, A260734, A276574, A277191, A277192, A277487, A277889, A277890.

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From Charles R Greathouse IV, Jul 19 2011
A277891(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + (k%2); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277891.txt", n, " ", A277891(n)));

(define (A277891 n) (let ((org_k (- (A000290 (+ 1 n)) 1))) (let loop ((k org_k) (s 0)) (if (and (< k org_k) (= 1 (A010052 (+ 1 k)))) s (loop (- k (A002828 k)) (+ s (A000035 k)))))))

a(n) + A277890(n) = A260734(n).

a(n) >= A277487(n).

A277191 Number of integers k in range [n^2, ((n+1)^2)-1] for which the least number of squares that add up to k (A002828) is odd.

Original entry on oeis.org

0, 2, 2, 4, 5, 5, 7, 7, 9, 10, 10, 13, 12, 15, 16, 15, 18, 19, 20, 20, 24, 21, 25, 26, 26, 27, 30, 30, 31, 33, 33, 33, 37, 37, 38, 40, 42, 41, 43, 44, 46, 46, 48, 47, 51, 50, 51, 55, 53, 56, 57, 58, 61, 61, 64, 61, 63, 66, 69, 65, 68, 71, 73, 70, 74, 75, 76, 77, 79, 79, 81, 85, 80, 86, 85, 85, 89, 90, 91, 91, 92, 95, 95, 98, 95, 100

Offset: 0

Antti Karttunen, Oct 04 2016

nonn

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

Cf. A000035, A000290, A002828, A277192.

After the initial zero, one more than A277193.

(define (A277191 n) (add (lambda (i) (A000035 (A002828 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)))))))

a(n) = Sum_{i=n^2 .. ((n+1)^2)-1} A000035(A002828(i)).
For all n >= 0, a(n) + A277192(n) = 2n+1.
For all n >= 1, a(n) = 1 + A277193(n).

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).

cp's OEIS Frontend

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

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A277891 a(n) = number of odd numbers encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A277191 Number of integers k in range [n^2, ((n+1)^2)-1] for which the least number of squares that add up to k (A002828) is odd.

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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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