A022544 -id:A022544 - cp's OEIS frontend

A260157 Smallest term of the first run of at least n consecutive integers which are not sums of 2 squares.

3, 6, 21, 21, 75, 91, 186, 378, 987, 987, 1494, 1494, 1494, 1494, 5166, 5166, 5166, 5166, 16110, 16869, 31658, 31658, 31658, 52394, 101350, 101350, 101350, 105573, 241883, 241883, 284003, 284003, 284003, 685542, 1437354, 1751297, 1853867, 1853867, 1853867, 1853867, 1853867, 1853867, 1853867, 1853867, 1853867, 1853867, 1853867

Offset: 1

Ivan Neretin, Nov 09 2015

nonn

			None of 21, 22, 23, and 24 is representable as a sum of two squares. Previous record of run length was 2, hence a(3)=a(4)=21.

Cf. A022544.

t = Select[Range[10^4], SquaresR[2, #] == 0 &]; SelectFirst[t, Function[n, ContainsAll[t, n + Range@ #]]] & /@ Range[0, 15] (* Michael De Vlieger, Nov 09 2015, Version 10.2 *)

A261604 a(1)=0. For n>1, a(n) = smallest number > a(n-1) such that, for all m,r

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 33, 35, 38, 39, 40, 42, 43, 44, 46, 47, 48, 51, 53, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 68, 69, 70, 71, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88

Offset: 1

Anders Hellström, Aug 25 2015

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Anders Hellström, Ruby program

A022544 is a subsequence.

Cf. A026468, A026466, A026467, A026469, A026470.

issumsq(n,r,s)=(r^2)+(s^2)==n
first(m)=my(v=vector(m), x, r, n, s); v[1]=0; for(n=2, m, v[n]=v[n-1]+1;until(x==1, for(r=1, n-1, for(s=1, n-1, if(issumsq(v[n],v[r],v[s]), v[n]++; x=0; break(2), x=1))))); v;

isA022544(n)=if(n%4==3, return(1)); my(f=factor(n)); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(1))); 0
search(v,x)=my(t=setsearch(v,x)); if(t, t, setsearch(v,x,1))
list(lim)=my(v=List([0,1]),t); for(n=3,lim, if(isA022544(n), listput(v,n); next); for(j=search(v,sqrtint((n-1)\2)+1),search(v,sqrtint(n)), if(issquare(n-v[j]^2, &t) && setsearch(v,t), next(2))); listput(v,n)); Set(v) \\ Charles R Greathouse IV, Sep 01 2015

a(n) ~ n, and in particular a(n) = n + O(n/sqrt(log n)). I do not know if this bound is tight. - Charles R Greathouse IV, Sep 01 2015

A264101 Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.

Original entry on oeis.org

23, 33, 47, 62, 63, 86, 118, 134, 138, 143, 158, 167, 188, 195, 203, 204, 209, 223, 230, 243, 248, 275, 283, 294, 318, 323, 348, 368, 383, 385, 395, 398, 408, 411, 413, 418, 419, 426, 437, 440, 448, 454, 467, 473, 476, 489, 492, 503, 508, 518, 523, 558, 563, 566, 572, 608

Offset: 1

Alex Ratushnyak, Nov 03 2015

nonn

Intersection of A014134, A020757, A022544.

			Since 22 = 16+6, because 16 is a square and 6 is a triangular number, 22 is not a term.
23 is a term because there is no representation as S+T or S1+S2 or T1+T2, where S, S1, S2 are squares, and T, T1, T2 are triangular numbers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A014134, A020757, A022544, A264118.

N:= 1000: # for terms <= N
S:= [seq(i^2,i=0..floor(sqrt(N)))]: nS:= nops(S):
T:= [seq(i*(i+1)/2, i=0..floor(sqrt(2*N)))]: nT:= nops(T):
sort(convert({$1..N} minus {seq(seq(S[i]+S[j], j=1..i),i=1..nS),
seq(seq(S[i]+T[j],i=1..nS),j=1..nT),
seq(seq(T[i]+T[j],j=1..i),i=1..nT)}, list)); # Robert Israel, May 19 2020

mx = 610; Complement[ Range@ mx, Union@ Flatten@ Table[{i^2 + j^2, i(i + 1)/2 + j^2, i(i + 1)/2 + j(j + 1)/2}, {i, 0, Sqrt[2 mx]}, {j, 0, Sqrt[2 mx]}]] (* Robert G. Wilson v, Nov 29 2015 *)

A297350 Start of record gaps between sums of two squares.

Original entry on oeis.org

0, 2, 5, 20, 74, 90, 185, 377, 986, 1493, 5165, 16109, 16868, 31657, 52393, 101349, 105572, 241882, 284002, 685541, 1437353, 1751296, 1853866, 5588305, 9565544, 13305524, 20875482, 67070173, 135628357, 192085714, 264428585, 345869506, 426063725, 434120338, 672657850

Offset: 1

Charles R Greathouse IV, Dec 28 2017

nonn

Numbers of the form A001481(i) such that the difference A001481(i+1)-A001481(i) reaches record values, where i is the index of a(n) in A001481. - Felix Fröhlich, Jan 09 2018

			20 = 4^2 + 2^2 and 25 = 5^2 + 0^2 are both sums of two squares, but none of 21, 22, 23, or 24 are, and no previous gap is as long as 25 - 20 = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..43

Cf. A001481, A022544, A357018.

Block[{s = Select[Range[0, 10^6], SquaresR[2, #] != 0 &], t}, t = Differences@ s; s[[First@ FirstPosition[t, #] ]] & /@ Union@ FoldList[Max, t]] (* Michael De Vlieger, Jan 09 2018 *)

is2(f)=for(i=if(f[1,1]==2,2,1),#f~, if(bitand(f[i,2],1)==1 && bitand(f[i,1],3)==3, return(0))); 1
print1(r=0); last=2; forfactored(n=last+1,10^9, if(!is2(n[2]), next); t=n[1]-last; if(t>r, r=t; print1(", "last)); last=n[1])

A343992 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are perfect.

Original entry on oeis.org

0, 1, 0, 1, 3, 0, 0, 6, 3, 20, 0, 0, 29, 0, 0, 56, 101, 108, 0, 392

Offset: 1

N. J. A. Sloane, May 06 2021

Curves of order n generated by folding morphisms are walks on the square grid, also coded by sequences (starting with D) of n-1 U's and D's starting with D, the Up and Down folds. These are also known as n-folds. In the square grid they uniquely correspond to folding morphisms, which are a special class of morphisms sigma on the alphabet {a,b,c,d}. (There is in particular the requirement that sigma(a) = ab...). Here the letters a,b,c, and d correspond to the four possible steps of the walk. A curve C = C1 of order n generates curves Cj of order n^j by the process of iterated folding. Iterated folding corresponds to iterates of the folding morphism. Grid-filling or plane-filling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj. Perfect means that four 90-degree rotated copies of the curves Cj started at the origin will pass exactly twice through all grid-points as j tends to infinity (except the origin itself).

It is a theorem that a(A022544(n)) = 0, and a(A001481(n)) > 0 for n>2.

			For n=2 one obtains Heighway's dragon curve, with folding morphism sigma: a -> ab, b -> cb, c -> cd, d -> ad (see A105500 or A246960).

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted and updated in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614.

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
F. M. Dekking, Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles, Theoretical Computer Science, volume 414, issue 1, January 2012, pages 20-37. Also arXiv:1011.5788 [math.CO], 2010-2011.

Cf. A296148, A343990, A343991.

Renamed and rewritten by Michel Dekking, Jun 03 2021

A356809 Fibonacci numbers which are not the sum of two squares.

Original entry on oeis.org

3, 21, 55, 987, 2584, 6765, 17711, 46368, 317811, 832040, 2178309, 5702887, 14930352, 102334155, 267914296, 701408733, 1836311903, 4807526976, 12586269025, 32951280099, 86267571272, 225851433717, 591286729879, 1548008755920, 10610209857723

Offset: 1

Ctibor O. Zizka, Aug 29 2022

nonn

			F(4) = 3; 3 != x^2 + y^2 as no positive integers x, y >= 0 are the solution of this Diophantine equation.

Chai Wah Wu, Table of n, a(n) for n = 1..472

Intersection of A000045 and A022544.

Cf. A001481, A022340, A124132, A124134, A236264.

Select[Fibonacci[Range[65]], SquaresR[2, #] == 0 &] (* Amiram Eldar, Aug 29 2022 *)

is(n)=if(n%4==3, return(1)); my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0; \\ A022544
lista(nn) = select(is, apply(fibonacci, [1..nn])); \\ Michel Marcus, Sep 04 2022

from itertools import islice
from sympy import factorint
def A356809_gen(): # generator of terms
    a, b = 1, 2
    while True:
        if any(p&3==3 and e&1 for p, e in factorint(a).items()):
            yield a
        a, b = b, a+b
A356809_list = list(islice(A356809_gen(),30)) # Chai Wah Wu, Jan 10 2023

A362961 a(n) = Sum_{b=0..floor(sqrt(n)), n-b^2 is square} b.

Original entry on oeis.org

1, 1, 0, 2, 3, 0, 0, 2, 3, 4, 0, 0, 5, 0, 0, 4, 5, 3, 0, 6, 0, 0, 0, 0, 12, 6, 0, 0, 7, 0, 0, 4, 0, 8, 0, 6, 7, 0, 0, 8, 9, 0, 0, 0, 9, 0, 0, 0, 7, 13, 0, 10, 9, 0, 0, 0, 0, 10, 0, 0, 11, 0, 0, 8, 20, 0, 0, 10, 0, 0, 0, 6, 11, 12, 0, 0, 0, 0, 0, 12, 9, 10, 0

Offset: 1

Darío Clavijo, May 10 2023

a(n) = 0 if n in A022544.

a(n) > 0 if n in A001481.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A022544, A001481.

Cf. A143574 (sum of b^2), A000925.

a[n_]:=Sum[b Boole[IntegerQ[Sqrt[n-b^2]]],{b,0,Floor[Sqrt[n]]}]; Array[a,83] (* Stefano Spezia, May 15 2023 *)

a(n) = sum(b=0, sqrtint(n), if (issquare(n-b^2), b)); \\ Michel Marcus, May 16 2023

from gmpy2 import *
a = lambda n: sum([b for b in range(0, isqrt(n) + 1) if is_square(n - (b*b))])
print([a(n) for n in range(1, 84)])

from sympy import divisors
from sympy.solvers.diophantine.diophantine import cornacchia
def A362961(n):
    c = 0
    for d in divisors(n):
        if (k:=d**2)>n:
            break
        q, r = divmod(n,k)
        if not r:
            c += sum(d*(a[0]+(a[1] if a[0]!=a[1] else 0)) for a in cornacchia(1,1,q) or [])
    return c # Chai Wah Wu, May 15 2023

A363051 a(n) = Sum_{b=0..floor(sqrt(n/2)), n-b^2 is square} b.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 1, 3, 0, 2, 0, 0, 0, 0, 3, 1, 0, 0, 2, 0, 0, 4, 0, 3, 0, 0, 1, 0, 0, 2, 4, 0, 0, 0, 3, 0, 0, 0, 0, 6, 0, 4, 2, 0, 0, 0, 0, 3, 0, 0, 5, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 6, 3, 5, 0, 0, 0, 0, 0, 4, 0, 1, 0

Offset: 1

Darío Clavijo, May 14 2023

nonn

a(n) = 0 if n in A022544.

a(n) > 0 if n in A001481.

Project Euler, Problem 273: Sum of Squares

Cf. A022544, A001481, A362961.

A363051 := proc(n)
    local x,a ;
    a := 0 ;
    for x from 1 do
        if x^2 > n/2 then
            return a;
        end if;
        if issqr(n-x^2) then
            a := a+x ;
        end if;
    end do:
end proc:
seq(A363051(n),n=1..100) ; # R. J. Mathar, Jan 31 2024

a[n_]:=Sum[b Boole[IntegerQ[Sqrt[n-b^2]]],{b,0,Floor[Sqrt[n/2]]}]; Array[a,83] (* Stefano Spezia, May 15 2023 *)

from gmpy2 import *
a = lambda n: sum([b for b in range(0, isqrt(n >> 1) + 1) if is_square(n - (b*b))])
print([a(n) for n in range(1, 84)])

from sympy.solvers.diophantine.diophantine import diop_DN
def A363051(n): return sum(min(a) for a in diop_DN(-1,n))>>1 # Chai Wah Wu, May 16 2023

A034026 Numbers that are primitively or imprimitively represented by x^2+y^2, but not both.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153

Offset: 1

N. J. A. Sloane

nonn

Cf. A001481, A008784, A022544, A034023-.

A071010 Sigma(k)/4 when k is not a sum of 2 squares.

Original entry on oeis.org

1, 3, 2, 3, 7, 6, 6, 5, 8, 9, 6, 15, 10, 14, 18, 8, 12, 12, 15, 14, 24, 11, 21, 18, 12, 31, 18, 30, 18, 30, 20, 15, 42, 24, 26, 36, 17, 24, 36, 18, 31, 35, 24, 42, 20, 21, 56, 33, 30, 45, 28, 42, 32, 36, 30, 63, 39, 54, 26, 48, 27, 70, 54, 38, 62, 60, 36, 45, 36, 90, 42, 56, 78

Offset: 1

Benoit Cloitre, May 19 2002

Conjecture : if n is not the sum of 2 squares sigma(n) == 0 (mod 4) (converse is not true : if sigma(n) == 0 (mod 4), n is sometimes the sum of 2 squares : sigma(65) = 84 == 0 (mod 4) but 65 = 49+16 is a sum of 2 squares).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A022544.

DivisorSigma[1, Select[Range[120], SquaresR[2, #] == 0 &]]/4  (* Amiram Eldar, May 13 2022 *)

for(n=0,200,if(sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1)))==0,print1(sigma(n)/4,",")))

a(n) = sigma(A022544(n))/4.

cp's OEIS Frontend

A260157 Smallest term of the first run of at least n consecutive integers which are not sums of 2 squares.

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A261604 a(1)=0. For n>1, a(n) = smallest number > a(n-1) such that, for all m,r

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A264101 Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.

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A297350 Start of record gaps between sums of two squares.

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A343992 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are perfect.

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A356809 Fibonacci numbers which are not the sum of two squares.

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A362961 a(n) = Sum_{b=0..floor(sqrt(n)), n-b^2 is square} b.

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A363051 a(n) = Sum_{b=0..floor(sqrt(n/2)), n-b^2 is square} b.

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