A025426 -id:A025426 - cp's OEIS frontend

A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.

2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, 160225, 1221025, 2442050, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 61051250, 5928325, 303460625, 53955078125, 35409725, 100140625, 1289367675781250

Offset: 1

Robert G. Wilson v

			a(0) = 1 as 1 is the least positive integer not expressible as the sum of two squared positives.
a(1) = 2 from 2 = 1^2 + 1^2.
a(2) = 50 from 50 = 1^2 + 7^2 = 5^2 + 5^2.

A. Beiler, Recreations in the Theory of Numbers, Dover, pp. 140-141.

T. D. Noe and Ray Chandler, Table of n, a(n) for n = 1..2178 (a(2179) exceeds 1000 digits).
C. Rivera, Puzzle 62
Eric Weisstein's World of Mathematics, Square Number
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A018825, A048610, A025284-A025293 (first entries).

See A000446, A124980 and A093195 for other versions.

Array[Block[{k = 1}, While[Length@ DeleteCases[PowersRepresentations[k, 2, 2], ?(! FreeQ[#, 0] &)] != #, k++]; k] &, 6] (* _Michael De Vlieger, Mar 31 2019 *)

b(k)=my(c=0);for(i=1,sqrtint(k\2),if(issquare(k-i^2),c+=1));c \\ A025426
for(n=1,10,k=1;while(k,if(b(k)==n,print1(k,", ");break);k+=1)) \\ Derek Orr, Mar 20 2019

a(n) = min(2*A018782(2n-1), A018782(2n), A018782(2n+1)).

Corrected and extended by Jud McCranie

Definition improved by several correspondents, Nov 12 2007

A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways.

Original entry on oeis.org

50, 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 325, 338, 340, 365, 370, 377, 410, 425, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 650, 680, 685, 689, 697, 725, 730, 740, 745, 754, 765

Offset: 1

N. J. A. Sloane

A025426(a(n)) > 1. - Reinhard Zumkeller, Aug 16 2011
For the question that is in the link AskNRICH Archive: It is easy to show that (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2 = (a*d + b*c)^2 + (a*c - b*d)^2. So terms of this sequence can be generated easily. 5 is the least number of the form a^2 + b^2 where a and b distinct positive integers and this is a list sequence. This is the why we observe that there are many terms which are divisible by 5. - Altug Alkan, May 16 2016
Square roots of square terms: {25, 50, 65, 75, 85, 100, 125, 130, 145, 150, 169, 170, 175, 185, 195, 200, 205, 221, 225, 250, 255, 260, 265, 275, 289, 290, 300, 305, ...}. They are also listed by A009177. - Michael De Vlieger, May 16 2016

			50 is a term since 1^2 + 7^2 and 5^2 + 5^2 equal 50.
25 is not a term since though 3^2 + 4^2 = 25, 25 is square, i.e., 0^2 + 5^2 = 25, leaving it with only one possible sum of 2 nonzero squares.
625 is a term since it is the sum of squares of {0,25}, {7,24}, and {15,20}.

Ming-Sun Li, Kathryn Robertson, Thomas J. Osler, Abdul Hassen, Christopher S. Simons and Marcus Wright, "On numbers equal to the sum of two squares in more than one way", Mathematics and Computer Education, 43 (2009), 102 - 108.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 125.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
AskNRICH Archive, Numbers expressible as the sum of 2 squares in more than one way
D. J. C. Mackay and S. Mahajan, Numbers that are Sums of Squares in Several Ways
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Subsequence of A001481. A subsequence is A025285 (2 ways).

Cf. A004431, A118882, A000404, A018825, A025284 (one way).

import Data.List (findIndices)
a007692 n = a007692_list !! (n-1)
a007692_list = findIndices (> 1) a025426_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range@ 800, Length@ Select[PowersRepresentations[#, 2, 2], First@ # != 0 &] > 1 &] (* Michael De Vlieger, May 16 2016 *)

isA007692(n)=nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb >= 2; \\ Altug Alkan, May 16 2016

is(n)=my(t); if(n<9, return(0)); for(k=sqrtint(n\2-1)+1,sqrtint(n-1), if(issquare(n-k^2)&&t++>1, return(1))); 0 \\ Charles R Greathouse IV, Jun 08 2016

A025284 Numbers that are the sum of 2 nonzero squares in exactly 1 way.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 72, 73, 74, 80, 82, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 128, 136, 137, 146, 148, 149, 153, 157, 160, 162, 164, 169, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212

Offset: 1

David W. Wilson

nonn

A025426(a(n)) = 1. - Reinhard Zumkeller, Aug 16 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Number
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A000404, A025426, A081324, A125022, A018825, A001481, A007692, A243148.

import Data.List (elemIndices)
a025284 n = a025284_list !! (n-1)
a025284_list = elemIndices 1 a025426_list
-- Reinhard Zumkeller, Aug 16 2011

selQ[n_] := Length[ Select[ PowersRepresentations[n, 2, 2], Times @@ # != 0 &]] == 1; Select[Range[300], selQ] (* Jean-François Alcover, Oct 03 2013 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 2], {n, 0, 300}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

A243148(a(n),2) = 1. - Alois P. Heinz, Feb 25 2019

A018825 Numbers that are not the sum of 2 nonzero squares.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 16, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 36, 38, 39, 42, 43, 44, 46, 47, 48, 49, 51, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 81, 83, 84, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110

Offset: 1

David W. Wilson

nonn

Cf. A022544, A081324, A000404 (complement), A004431.

import Data.List (elemIndices)
a018825 n = a018825_list !! (n-1)
a018825_list = tail $ elemIndices 0 a025426_list
-- Reinhard Zumkeller, Aug 16 2011

isA000404 := proc(n)
    local x,y ;
    for x from 1 do
        if x^2> n then
            return false;
        end if;
        for y from 1 do
            if x^2+y^2 > n then
                break;
            elif x^2+y^2 = n then
                return true;
            end if;
        end do:
    end do:
end proc:
A018825 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if not isA000404(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A018825(n),n=1..30) ; # R. J. Mathar, Jul 28 2014

q=13;q2=q^2+1;lst={};Do[Do[z=a^2+b^2;If[z<=q2,AppendTo[lst,z]],{b,a,1,-1}],{a,q}];lst; u=Union@lst;Complement[Range[q^2],u] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

is(n)=my(f=factor(n), t=prod(i=1,#f~, if(f[i,1]%4==1, f[i,2]+1, if(f[i,2]%2 && f[i,1]>2, 0, 1)))); if(t!=1, return(!t)); for(k=sqrtint((n-1)\2)+1, sqrtint(n-1), if(issquare(n-k^2), return(0))); 1 \\ Charles R Greathouse IV, Sep 02 2015

A025426(a(n)) = 0; A063725(a(n)) = 0. - Reinhard Zumkeller, Aug 16 2011

A025285 Numbers that are the sum of 2 nonzero squares in exactly 2 ways.

Original entry on oeis.org

50, 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 338, 340, 365, 370, 377, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 800, 820

Offset: 1

David W. Wilson

nonn

Order and signs don't count. E.g. 50 = 5^2+5^2 = 7^2+1^2 (= (-5)^2+5^2, but that doesn't count as different).
A131574 is a subsequence. - Zak Seidov, Jan 31 2014
A025426(a(n)) = 2. - Reinhard Zumkeller, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Square Number.
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A025284, A025286, A025287, A025288, A025289, A025290, A025291, A025292, A025293, A025426.

Cf. A007692.

a025285 n = a025285_list !! (n-1)
a025285_list = filter ((== 2) . a025426) [1..]
-- Reinhard Zumkeller, Feb 26 2015

selQ[n_] := Length[ Select[ PowersRepresentations[n, 2, 2], Times @@ # != 0 &]] == 2; Select[Range[1000], selQ] (* Jean-François Alcover, Oct 03 2013 *)

is(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n-1), issquare(n-k^2))==2 \\ Charles R Greathouse IV, May 24 2016

is(n)=my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)==4 \\ Charles R Greathouse IV, May 24 2016

a(n) >= A007692(n) with equality only for n <= 16. - Alois P. Heinz, Mar 23 2023

A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

Original entry on oeis.org

3, 9, 17, 19, 33, 35, 51, 73, 81, 99, 105, 129, 145, 147, 161, 163, 179, 195, 201, 233, 243, 273, 289, 291, 297, 339, 361, 387, 393, 451, 465, 467, 483, 489, 513, 521, 577, 579, 585, 611, 627, 649, 675, 721, 723, 739, 777, 801, 809, 819, 849, 883, 899, 915

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Analogous solutions exist for the sum of two identical squares z^2-1 = 2.r^2 (e.g. 99^2-1 = 2.70^2). Values of 'z' are the terms in sequence A001541, values of 'r' are the terms in sequence A001542.
Looking at a^2 + b^2 = c^2 - 1 modulo 4, we must have a and b even and c odd. Taking a = 2u, b = 2v and c = 2w - 1 and simplifying, we get u^2 + v^2 = w(w+1). - Franklin T. Adams-Watters, May 19 2008
If n is in this sequence, then so is n^(2^k), for all k >= 0. - Altug Alkan, Apr 13 2016

			E.g. 51^2 - 1 = 10^2 + 50^2 = 22^2 + 46^2 = 34^2 + 38^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
E.-B. Escott, Query 2521, L'Intermédiaire des Mathématiciens, 10 (1903), 285. [Contains errors]
Index entries for sequences related to sums of squares

Cf. A050796, A050797, A001541, A001542, A001333.

Cf. A140612, A002378.

t={}; Do[i=c=1; While[iJayanta Basu, Jun 01 2013 *)
Select[Range@ 1000, Length[PowersRepresentations[#^2 - 1, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 13 2016 *)

select( {is_A050795(n)=#qfbsolve(Qfb(1,0,1),n^2-1,2)}, [1..999]) \\ M. F. Hasler, Mar 07 2022

from itertools import islice, count
from sympy import factorint
def A050795_gen(startvalue=2): # generator of terms >= startvalue
    for k in count(max(startvalue,2)):
        if all(map(lambda d: d[0] % 4 != 3 or d[1] % 2 == 0, factorint(k**2-1).items())):
            yield k
A050795_list = list(islice(A050795_gen(),20)) # Chai Wah Wu, Mar 07 2022

a(n) = 2*A140612(n) + 1. - Franklin T. Adams-Watters, May 19 2008

{k : A025426(k^2-1)>0}. - R. J. Mathar, Mar 07 2022

A025455 a(n) is the number of partitions of n into 2 positive cubes.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

In other words, number of solutions to the equation x^3 + y^3 = n with x >= y > 0. - Antti Karttunen, Aug 28 2017

The first term > 1 is a(1729) = 2. - Michel Marcus, Apr 23 2019

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to sums of cubes

Cf. A025456, A025468, A003108, A003325, A000578, A048766, A001235 (two or more ways, positions where a(n) > 1).

Cf. also A025426, A216284.

Table[Count[IntegerPartitions[n,{2}],?(AllTrue[Surd[#,3],IntegerQ]&)],{n,0,110}] (* _Harvey P. Dale, Nov 23 2022 *)

(define (A025455 n) (let loop ((x (A048766 n)) (s 0)) (let* ((x3 (A000578 x)) (y3 (- n x3))) (if (< x3 y3) s (loop (- x 1) (+ s (if (and (> y3 0) (= (A000578 (A048766 y3)) y3)) 1 0))))))) ;; Antti Karttunen, Aug 28 2017

If a(n) > 0 then A025456(n + k^3) > 0 for k>0; a(A113958(n)) > 0; a(A003325(n)) > 0. - Reinhard Zumkeller, Jun 03 2006
a(n) >= A025468(n). - Antti Karttunen, Aug 28 2017
a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Secondary offset added by Antti Karttunen, Aug 28 2017

Secondary offset corrected by Michel Marcus, Apr 23 2019

A084888 Number of partitions of n^3 into two squares>0.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 8, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 2

Offset: 0

Reinhard Zumkeller, Jun 10 2003

nonn

a(A050804(n)) = 1.

			n=100: 100^3 = 1000000 = 960^2 + 280^2 = 936^2 + 352^2 = 800^2 + 600^2, therefore a(100)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Diophantine Equation: 2nd Powers.
Reinhard Zumkeller, Illustration for A084888 and A000404

Cf. A000404, A063725, A000578, A000290.

a084888 = a025426 . a000578  -- Reinhard Zumkeller, Jul 18 2012

a(n)=my(f=factor(n^3)); (prod(i=1,#f~,if(f[i,1]%4==1,f[i,2]+1,f[i,2]%2==0||f[i,1]<3))-issquare(n)+1)\2 \\ Charles R Greathouse IV, May 18 2016

from math import prod
from sympy import factorint
def A084888(n): return ((m:=prod(1 if p==2 else (3*e+1 if p&3==1 else (3*e+1)&1) for p, e in factorint(n).items()))+((((~n**3 & n**3-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1 # Chai Wah Wu, May 17 2023

a(n) = A025426(A000578(n)).

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

A321437 Expansion of Product_{1 <= i <= j} 1/(1 - x^(i^2 + j^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 1, 4, 3, 4, 5, 5, 6, 9, 6, 13, 8, 14, 13, 15, 19, 21, 21, 29, 24, 38, 32, 42, 44, 51, 56, 65, 65, 83, 79, 102, 99, 120, 125, 144, 154, 176, 183, 213, 219, 262, 266, 311, 322, 369, 392, 437, 465, 526, 550, 635, 650, 747, 781, 871, 937

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A321436.

Cf. A025426, A321431, A321435.

G.f.: Product_{k>0} 1/(1 - x^k)^A025426(k).

cp's OEIS Frontend

A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.

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A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways.

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A025284 Numbers that are the sum of 2 nonzero squares in exactly 1 way.

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A018825 Numbers that are not the sum of 2 nonzero squares.

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A025285 Numbers that are the sum of 2 nonzero squares in exactly 2 ways.

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A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

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A025455 a(n) is the number of partitions of n into 2 positive cubes.

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