A102548 -id:A102548 - cp's OEIS frontend

A229062 1 if n is representable as sum of two nonnegative squares, otherwise 0.

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1

Offset: 0

Ralf Stephan, Sep 17 2013

Characteristic function of A001481.
a(n) = 1 if A000161(n) > 0.
a(A022544(n)) = 0.
Multiplicative because A002654 is. - Andrew Howroyd, Aug 01 2018
For positive n, m = 2*a(n) + 1 is the smallest positive integer such that m * n is not a sum of two squares. - Peter Schorn, Dec 29 2023

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions

Cf. A002654, A004018, A070176. Partial sums are in A102548.

Cf. A000161, A022544.

Join[{1},Table[If[SquaresR[2,n]>1,1,0],{n,120}]] (* Harvey P. Dale, Aug 25 2017 *)

a(n)=my(f=0); my(r=sqrtint(n)); forstep(i=r, 1, -1, if(issquare(n-i*i), f=1; break)); f

a(n)=if(0==n,1,(sumdiv(n, d,(d%4==1) - (d%4==3)) > 0)); \\ Andrew Howroyd, Aug 01 2018, the check for 0-argument added by Antti Karttunen, Apr 22 2022

from sympy import factorint
def A229062(n): return int(all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items())) # Chai Wah Wu, Jun 28 2022

a(n) = min{1, A004018(n)}. - N. J. A. Sloane, Jan 11 2020

A160663 Number of distinct sums that one can obtain by adding two squares among the n first ones.

Original entry on oeis.org

2, 5, 9, 14, 19, 26, 33, 41, 50, 60, 70, 82, 93, 105, 119, 134, 147, 164, 179, 197, 215, 234, 251, 272, 293, 314, 336, 359, 381, 407, 430, 456, 483, 507, 535, 566, 594, 623, 652, 686, 714, 748, 780, 812, 849, 883, 918, 956, 992, 1030, 1068, 1107, 1141, 1181

Offset: 1

Romain CARRE (romain.carre.2008(AT)enseirb.fr), May 22 2009

nonn

Essentially the same as A047800: a(n) = A047800(n) - 1.
Let A be the set of the n first squares (1,4,9,...,n^2). Let A+A be the corresponding sumset (= {a,b,a+b where (a,b) in A^2}). That very sequence describes the number of elements of A+A, relatively to n.
a(n-1) is the number of distinct positive distances on an n X n pegboard. What is its asymptotic growth? Can it be efficiently computed for large n? - Charles R Greathouse IV, Jun 13 2013
An upper bound is a(n) <= A102548(2n^2) << n^2/log n. - Charles R Greathouse IV, Jan 16 2023

			For n = 3, A = {1,4,9}, A+A = {1,4,9} U {2,5,10,8,13,18} thus A+A = {1,2,4,5,8,9,10,13,18}, and hence card(A+A) = 9; a(3) = 9.

Melvyn B. Nathanson (1996). "Additive Number Theory: the Classical Bases" Graduate Texts in Mathematics. 164. Springer-Verlag. p. 192. ISBN 0-387-94656-X.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
L. G. Schnirelmann, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933) 649-690.
L. G. Schnirelmann, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933) 649-690. doi:10.1007/BF01448914.
Samuel S. Wagstaff, Jr., The Schnirelmann density of the sums of three squares, Proc. Amer. Math. Soc. 52 (1975), 1-7.
Wikipedia, Additive number theory
Wikipedia, Schnirelmann density
Wikipedia, Edmund Landau

a:= proc(n) local A, i, j; A:= [i^2$i=1..n]; nops([{A[], seq (seq (A[i]+A[j], j=1..i), i=1..nops(A))}[]]) end: seq (a(n), n=1..60); # Alois P. Heinz, Jun 16 2009

a[n_] := (Table[i^2 + j^2, {i, 0, n}, {j, i, n}] // Flatten // Union // Length) - 1; Array[a, 60] (* Jean-François Alcover, May 25 2018 *)

a(n)=n++; #vecsort(vector(n^2,i,((i-1)\n)^2+((i-1)%n)^2),,8)-1 \\ Charles R Greathouse IV, Jun 13 2013

a(n)=my(u=vector(n,i,i^2),v=List(u)); for(i=1,n, for(j=1,i, listput(v,u[i]+u[j]))); u=0; #Set(v) \\ Charles R Greathouse IV, Nov 18 2022

first(n)=my(v=vector(n),u=[]); for(k=1,n, my(k2=k^2,w=vector(k,i,i^2+k2)); w=setunion(w,[k2]); u=setunion(u,w); v[k]=#u); v \\ Charles R Greathouse IV, Nov 18 2022

def a(n):
    SUM, SQR = set(), set(x**2 for x in range(1, n + 1))
    for i in SQR:
        SUM.add(i)
        for j in SQR: SUM.add(i + j)
    return len(SUM)
# Romain CARRE (romain.carre.2008(AT)enseirb.fr), Apr 16 2010

a(n) = card(A+A) where A={k^2} k=1..n and A+A = {a,b,a+b where (a,b) in A^2}.
Trivially 2n <= a(n) <= n(n+1)/2. - Charles R Greathouse IV, Oct 30 2015
a(n) << n^2/sqrt(log n) [see A000404]. - Charles R Greathouse IV, Oct 30 2015

More terms from Alois P. Heinz, Jun 16 2009

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A229062 1 if n is representable as sum of two nonnegative squares, otherwise 0.

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A160663 Number of distinct sums that one can obtain by adding two squares among the n first ones.

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