A000161 -id:A000161 - 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

A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

Original entry on oeis.org

6, 21, 36, 55, 66, 91, 120, 136, 171, 210, 231, 276, 351, 378, 406, 496, 561, 666, 703, 741, 820, 861, 946, 990, 1035, 1081, 1176, 1225, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2016, 2080, 2211, 2278, 2346, 2556, 2701, 2775, 2850

Offset: 1

Jon Perry, Jan 13 2004

Intersection of triangular numbers with sumset of triangular numbers. Triangular number analog of what for squares is {A057100(n)^2} = {A009000(n)^2}. {A000217} INTERSECT {A000217 + A000217}. - Jonathan Vos Post, Mar 09 2007

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			Generally, A000217(A000217(n)) = A000217(A000217(n)-1) + A000217(n) and so is automatically included. These are 6=T(3), 21=T(6), 55=T(10), etc. Other solutions occur when a partial sum from x to y is triangular, e.g., 15 + 16 + 17 + 18 = 66 = T(11), so T(14) + T(11) = T(18). This particular example arises since 10+4k is triangular (at k=14, 10 + 4k = 66), and we therefore have a solution.
All other solutions occur when 3+2k, 6+3k, 10+4k, etc. -- in general, T(j) + j*k -- is triangular.

Michael S. Branicky, Table of n, a(n) for n = 1..11915

Cf. A000217, A012132, A051533, A057100.

trn[i_]:=Module[{trnos=Accumulate[Range[i]],t2s},t2s=Union[Total/@ Tuples[ trnos,2]];Intersection[trnos,t2s]] (* Harvey P. Dale, Nov 08 2011 *)
Select[Range[75], ! PrimeQ[#^2 + (# + 1)^2] &] /. Integer_ -> (Integer^2 + Integer)/2 (* Arkadiusz Wesolowski, Dec 03 2015 *)

t(i) = i*(i+1)/2;
{ v=vector(100,i,t(i)); y=vector(100); c=0; for (i=1,30, for (j=i,30, x=t(i)+t(j); f=0; for (k=1,100,if (x==v[k],f=1;break)); if (f==1,y[c++ ]=x))); select(x->(x>0), vecsort(y,,8)) } \\ slightly edited by Michel Marcus, Apr 15 2021

lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); for (k=1, n-1, if (ispolygonal(t - k*(k+1)/2, 3), print1(t, ", "); break;)););} \\ Michel Marcus, Apr 15 2021

from itertools import count, takewhile
def aupto(lim):
    t = list(takewhile(lambda x: x<=lim, (i*(i+1)//2 for i in count(1))))
    s = set(a+b for i, a in enumerate(t) for b in t[i:])
    return sorted(s & set(t))
print(aupto(3000)) # Michael S. Branicky, Jun 21 2021

Triangular number m is in this sequence iff A000161(4*m+1)>1 or, alternatively, A083025(4*m+1)>1. - Max Alekseyev, Oct 24 2008

a(n) = A000217(A012132(n)). - Ivan N. Ianakiev, Jan 17 2013

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net) and David Wasserman, Sep 23 2005

A124982 Nonprime numbers with a unique partition as a sum of 2 squares x^2 + y^2.

Original entry on oeis.org

0, 1, 4, 8, 9, 10, 16, 18, 20, 26, 32, 34, 36, 40, 45, 49, 52, 58, 64, 68, 72, 74, 80, 81, 82, 90, 98, 104, 106, 116, 117, 121, 122, 128, 136, 144, 146, 148, 153, 160, 162, 164, 178, 180, 194, 196, 202, 208, 212, 218, 226, 232, 234, 242, 244, 245, 256, 261, 272, 274

Offset: 1

Artur Jasinski, Nov 16 2006

nonn

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

Cf. A000161, A002145.

Select[Range[0, 300], !PrimeQ[#] && Length @ PowersRepresentations[#, 2, 2] == 1 &] (* Amiram Eldar, Mar 12 2020 *)

A000161(n)={ local(cnt=0,y2) ; for(x=0,floor(sqrt(n)), y2=n-x^2 ; if( y2>=x^2 && issquare(y2), cnt++ ; ) ; ) ; return(cnt) ; } isA124982(n)= { if( isprime(n), return(0), if(A000161(n)==1, return(1), return(0) ) ) ; } { for(n=0,300, if( isA124982(n), print1(n,", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 29 2006

A000161(a(n)) = 1. - R. J. Mathar, Nov 29 2006

Corrected and extended by R. J. Mathar, Nov 29 2006

A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 3, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 1, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0

Offset: 1

V. Raman, Sep 03 2012

nonn

Records occur at 1, 9, 81, 297, 891, 1683, 5049, 15147, 31977, ... - Antti Karttunen, Aug 23 2017

			For n = 9, there are two solutions: 9 = 9^2 + 2*(0^2) = 1^2 + 2*(2^2), thus a(9) = 2.
For n = 81, there are three solutions: 81 = 9^2 + 2*(0^2) = 3^2 + 2*(6^2) = 7^2 + 2*(4^2), thus a(81) = 3.
For n = 65536, there is one solution: 65536 = 256^2 + 2*(0^2) = 65536 + 0, thus a(65536) = 1.
For n = 65537, there is one solution: 65537 = 255^2 + 2*(16^2) = 65205 + 512, thus a(65537) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A092573, A119395, A000161, A025426, A216283.

Cf. A002479 (positions of nonzeros), A097700 (of zeros).

r[n_] := Reduce[x >= 0 && y >= 0 && x^2 + 2 y^2 == n, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2017 *)

(define (A216282 n) (cond ((< n 2) 1) (else (let loop ((k (- (A000196 n) (modulo (- n (A000196 n)) 2))) (s 0)) (if (< k 0) s (let ((x (/ (- n (* k k)) 2))) (loop (- k 2) (+ s (A010052 x))))))))) ;; Antti Karttunen, Aug 23 2017

Examples from Antti Karttunen, Aug 23 2017

A050797 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in exactly one way.

Original entry on oeis.org

3, 9, 17, 19, 33, 35, 73, 145, 161, 163, 195, 243, 393, 483, 513, 721, 723, 1153, 1763, 2177, 2305, 2593, 4803, 5185, 5833, 6273, 6963, 7057, 7395, 8713, 9523, 9603, 10083, 12483, 13923, 14113, 15875, 17425, 17673, 19043, 20737

Offset: 1

Patrick De Geest, Sep 15 1999

If the definition were changed from "nonzero squares" to "nonnegative squares", there would be just one additional term, 1. - T. D. Noe, May 27 2008

			E.g. 393^2 - 1 = 28^2 + 392^2 only.

T. D. Noe, Table of n, a(n) for n=1..300
Eric Weisstein, MathWorld: Sum of Squares Function
Index entries for sequences related to sums of squares

Cf. A050798, A050795.

Cf. A000161

twoSquaresQ[ n_] := (r = Reduce [0 < a <= b && n^2 - 1 == a^2 + b^2, {a, b}, Integers]; Head[r] === And); Select[ Range[21000], twoSquaresQ] (* Jean-François Alcover, Oct 10 2011 *)

More terms from James Sellers

A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

Original entry on oeis.org

1, 25, 325, 1105, 4225, 5525, 203125, 27625, 71825, 138125, 2640625, 160225, 17850625, 1221025, 1795625, 801125, 1650390625, 2082925, 49591064453125, 4005625, 44890625, 2158203125, 30525625, 5928325, 303460625, 53955078125, 35409725, 100140625

Offset: 1

Artur Jasinski, Nov 15 2006

nonn

The number must be strictly positive, but one of the squares may be zero, as we see from a(1) = 1 = 1^2 + 0^2 and a(2) = 25 = 3^2 + 4^2 = 5^0 + 0^2. - M. F. Hasler, Jul 07 2024

			a(3) = 325 is decomposable in 3 ways: 15^2 + 10^2 = 17^2 + 6^2 = 18^2 + 1^2.

Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).

Cf. A002144, A018782, A054994, A118882.

See A016032, A000446 and A093195 for other versions.

A124980(n)={for(a=1, oo, A000161(a)==n && return(a))} \\ R. J. Mathar, Nov 29 2006, edited by M. F. Hasler, Jul 07 2024

PD(n, L=n, D=Vecrev(divisors(n)[^1])) = { if(n>1, concat(vector(#D, i, if(D[i] > L, [], D[i] < n, [concat(D[i], P) | P <- PD(n/D[i], D[i])], [[n]]))), [[]])}
apply( {A124980(n)=vecmin([prod(i=1, #a, A002144(i)^(a[i]-1)) | a<-concat([PD(n*2,n), PD(n*2-1)])])}, [1..44]) \\ M. F. Hasler, Jul 07 2024

from sympy import divisors, isprime, prod
def PD(n, L=None): return [[]] if n==1 else [
    [d]+P for d in divisors(n)[:0:-1] if d <= (L or n) for P in PD(n//d, d)]
A2144=lambda upto=999: filter(isprime, range(5, upto, 4))
def A124980(n):
    return min(prod(a**(f-1) for a,f in zip(A2144(), P))
               for P in PD(n*2, n)+PD(n*2-1)) # M. F. Hasler, Jul 07 2024

a(n) = A000446(n), n > 1. - R. J. Mathar, Jun 15 2008
a(n) = min(A018782(2n-1), A018782(2n)).
a(n) = min { k > 0 | A000161(k) = n }. - M. F. Hasler, Jul 07 2024

More terms from R. J. Mathar, Nov 29 2006

Edited and extended by Ray Chandler, Jan 07 2012

A143574 Sum of all distinct squares occurring when partitioning n into two squares.

Original entry on oeis.org

0, 1, 1, 0, 4, 5, 0, 0, 4, 9, 10, 0, 0, 13, 0, 0, 16, 17, 9, 0, 20, 0, 0, 0, 0, 50, 26, 0, 0, 29, 0, 0, 16, 0, 34, 0, 36, 37, 0, 0, 40, 41, 0, 0, 0, 45, 0, 0, 0, 49, 75, 0, 52, 53, 0, 0, 0, 0, 58, 0, 0, 61, 0, 0, 64, 130, 0, 0, 68, 0, 0, 0, 36, 73, 74, 0, 0, 0, 0, 0, 80, 81, 82, 0, 0, 170, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Aug 24 2008

nonn

For n > 0: a(n) = 0 iff A000161(n) = 0: a(A022544(n)) = 0;

A143575 gives numbers m such that a(m) = m.

			A000161(25)=#{5^2+0^2,4^2+3^2}=2: a(25)=25+0+16+9=50;
A000161(26)=#{5^2+1^2}=1: a(16)=25+1=26;
A000161(49)=#{7^2+0^2}=1: a(49)=49+0=49;
A000161(50)=#{7^2+1^2,5^2+5^2}=2: a(50)=49+1+25=75;
A000161(2600)=#{50^2+10^2,46^2+22^2,38^2+34^2}=3: a(2600)=2500+100+2116+484+1444+1156=7800;
A000161(2601)=#{51^2+0^2,45^2+24^2}=2: a(2601)=2601+0+12025+576=5202;
A000161(2602)=#{51^2+1^2}=1: a(2602)=2601+1=2602.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000161, A002654, A010052, A022544, A143575.

a(n) = sum(k=1, n, if (issquare(k) && issquare(n-k), k)); \\ Michel Marcus, May 16 2023

from sympy import divisors
from sympy.solvers.diophantine.diophantine import cornacchia
def A143574(n):
    c = 0
    for d in divisors(n):
        if (k:=d**2)>n:
            break
        q, r = divmod(n,k)
        if not r:
            c += sum(k*(a[0]**2+(a[1]**2 if a[0]!=a[1] else 0)) for a in cornacchia(1,1,q) or [])
    return c # Chai Wah Wu, May 15 2023

a(n) = Sum_{k=1..n} k*A010052(k)*A010052(n-k). [Reinhard Zumkeller, Sep 27 2008]

A007511 a(n) is the smallest number greater than a(n-1) that is expressible as the sum of two squares in more ways than a(n-1).

Original entry on oeis.org

2, 50, 325, 1105, 5525, 27625, 71825, 138125, 160225, 801125, 2082925, 4005625, 5928325, 29641625, 77068225, 148208125, 243061325, 1215306625, 3159797225, 6076533125, 12882250225, 53716552825, 64411251125, 167469252925, 322056255625, 785817263725

Offset: 1

Gabriel Cunningham (gcasey(AT)mit.edu), Feb 29 2004

nonn

Sequence provides the locations of records in A025426 (nonzero squares), rather than in A000161 (definition of squares includes zeros). - R. J. Mathar, Jun 06 2007

J. Meeus, Note

Cf. A048610.

a(12)-a(18) from Donovan Johnson, Sep 03 2008
a(19)-a(24) from Donovan Johnson, Jul 01 2009
a(25)-a(26) from Donovan Johnson, Aug 30 2011

A046711 From the Bruck-Ryser theorem: numbers n == 1 or 2 (mod 4) which are also the sum of 2 squares.

Original entry on oeis.org

1, 2, 5, 9, 10, 13, 17, 18, 25, 26, 29, 34, 37, 41, 45, 49, 50, 53, 58, 61, 65, 73, 74, 81, 82, 85, 89, 90, 97, 98, 101, 106, 109, 113, 117, 121, 122, 125, 130, 137, 145, 146, 149, 153, 157, 162, 169, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 225

Offset: 1

N. J. A. Sloane

Intersection of A001481 and A042963; A000161(a(n)) > 0. - Reinhard Zumkeller, Feb 14 2012

M. Hall, Jr., Combinatorial Theory, Theorem 12.3.2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. H. Bruck and H. J. Ryser, The nonexistence of certain projective planes, Canad. J. Math., 1 (1949), 88-93.
Index entries for sequences related to sums of squares

Cf. A000161, A001481, A042963.

a046711 n = a046711_list !! (n-1)
a046711_list = [x | x <- a042963_list, a000161 x > 0]
-- Reinhard Zumkeller, Aug 16 2011

max = 225; Flatten[ Table[ a^2 + b^2, {a, 0, Sqrt[max]}, {b, a, Sqrt[max - a^2]}], 1] // Union // Select[#, (1 <= Mod[#, 4] <= 2)& ]& (* Jean-François Alcover, Sep 13 2012 *)
With[{max=15},Select[Select[Total/@Tuples[Range[0,max]^2,2], MemberQ[ {1,2}, Mod[ #,4]]&]//Union,#<=max^2&]] (* Harvey P. Dale, Jan 14 2017 *)

from itertools import count, islice
from sympy import factorint
def A046711_gen(): # generator of terms
    return filter(lambda n:0 < n & 3 < 3 and all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A046711_list = list(islice(A046711_gen(),30)) # Chai Wah Wu, Jun 28 2022

More terms from James Sellers

A050798 Numbers n such that m = n^2 + 1 is expressible as the sum of two nonzero squares in exactly two ways.

Original entry on oeis.org

1, 7, 8, 12, 13, 17, 21, 22, 23, 27, 28, 30, 31, 33, 34, 37, 41, 42, 44, 46, 48, 50, 52, 53, 55, 58, 60, 62, 63, 64, 67, 75, 76, 77, 78, 80, 81, 86, 87, 88, 89, 91, 92, 96, 97, 100, 102, 103, 104, 105, 106, 108, 109, 111, 113, 114, 115, 119, 125, 127, 129, 135, 136

Offset: 1

Patrick De Geest, Sep 15 1999

Of course m = n^2 + 1 is the sum of two squares, by definition. Here there should be just one other way to write m as a different sum of two squares.

Let p and q be primes of the form 1+4k. Then n^2+1 must be pq or 2pq. - T. D. Noe, May 27 2008

			E.g., 111^2 + 1 = 21^2 + 109^2 only.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein, MathWorld: Sum of Squares Function
Index entries for sequences related to sums of squares

Cf. A000161, A050797, A050796.

ok[1] = True; ok[n_] := Length[ {ToRules[ Reduce[ 1 < x <= y && n^2 + 1 == x^2 + y^2, {x, y}, Integers] ] } ] == 1; Select[ Range[136], ok] (* Jean-François Alcover, Feb 16 2012 *)

Better definition from T. D. Noe, May 27 2008

cp's OEIS Frontend

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

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A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

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A124982 Nonprime numbers with a unique partition as a sum of 2 squares x^2 + y^2.

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A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

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A050797 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in exactly one way.

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A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

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PARI

PARI

Python

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Extensions

A143574 Sum of all distinct squares occurring when partitioning n into two squares.

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