A001983 -id:A001983 - cp's OEIS frontend

A003995 Sum of (any number of) distinct squares: of form r^2 + s^2 + t^2 + ... with 0 <= r < s < t < ...

0, 1, 4, 5, 9, 10, 13, 14, 16, 17, 20, 21, 25, 26, 29, 30, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001983, A033461, A008935. Complement of A001422.

Cf. A000290; subsequences: A004431, A004432, A004433, A004434, A224981, A224982, A224983.

a003995 n = a003995_list !! (n-1)
a003995_list = filter (p a000290_list) [0..]
   where p (q:qs) m = m == 0 || q <= m && (p qs (m - q) || p qs m)
-- Reinhard Zumkeller, Apr 22 2013

lim = 10; s = {0}; Do[s = Union[s, s + n^2], {n, lim}]; Select[s, 0 <= # <= lim^2 &] (* T. D. Noe, Jul 10 2012 *)

a(n)=if(n<1,0,n=a(n-1); until(polcoeff(prod(k=1,sqrt(n),1+x^k^2), n), n++); n)

Exponents in expansion of (1+x)*(1+x^4)*(1+x^9)*(1+x^16)*(1+x^25)*(1+x^36)*(1+x^49)*(1+x^64)*(1+x^81)*(1+x^100)*(1+x^121)*(1+x^144)*...

For n > 98, a(n) = n + 30. - Charles R Greathouse IV, Sep 02 2011 (This implies a(n+2) = 2*a(n+1)-a(n) for n > 98.)

A025435 Number of partitions of n into 2 distinct squares.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

a(A004435(n)) = 0; a(A001983(n)) > 0. - Reinhard Zumkeller, Dec 20 2013

			G.f. = x + x^4 + x^5 + x^9 + x^10 + x^13 + x^16 + x^17 + x^20 + 2*x^25 + ...

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

Cf. A010052, A000290, A000161, A025441.

a025435 0 = 0
a025435 n = a010052 n + sum
   (map (a010052 . (n -)) $ takeWhile (< n `div` 2) $ tail a000290_list)
-- Reinhard Zumkeller, Dec 20 2013

A025435 := proc(n)
    local i, j, ans;
    ans := 0;
    for i from 0 to n do
        for j from i+1 to n do
            if i^2+j^2=n then
                ans := ans+1
            fi
        end do
    end do;
    ans ;
end proc: # R. J. Mathar, Aug 04 2018

a[ n_] := If[ n < 0, 0, Sum[ Boole[ n == i^2 + j^2], {i, Sqrt[n]}, {j, 0, i - 1}]]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := Length@ PowersRepresentations[ n, 2, 2] - Boole @ IntegerQ @ Sqrt[2 n]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := SeriesCoefficient[ With[ {f = (EllipticTheta[ 3, 0, x] + 1)/2, g = (EllipticTheta[ 3, 0, x^2] + 1)/2}, f f - g] / 2, {x, 0, n}]; (* Michael Somos, Jun 24 2015 *)

{a(n) = if( n<0, 0, sum(i=1, sqrtint(n), sum(j=0, i-1, n == i^2 + j^2)))}; /* Michael Somos, Jun 24 2015 */

A025435(n)=sum(k=sqrtint((n-1+!n)\2)+1, sqrtint(n), issquare(n-k^2))-issquare(n/2) \\ or A000161(n)-issquare(n/2). - M. F. Hasler, Aug 05 2018

from math import prod
from sympy import factorint
def A025435(n):
    f = factorint(n)
    return int(not any(e&1 for p,e in f.items() if p>2))*(1-((f.get(2,0)&1)<<1)) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 0 # Chai Wah Wu, Sep 08 2022

a(n) = A000161(n) - A010052(2*n). - M. F. Hasler, Aug 05 2018

a(n) = Sum_{i=1..n} c(i) * c(2*n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020

A052199 Numbers that are expressible as the sum of 2 distinct positive squares in more ways than any smaller number.

Original entry on oeis.org

1, 5, 65, 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

Jud McCranie, Jan 28 2000

nonn

			65 = 1^2 + 8^2 = 4^2 + 7^2, the smallest expressible in two ways, so 65 is a term.

Donald S. McDonald, Postings to sci.math newsgroup, Feb 21, 1995 and Dec 04, 1995.

Ray Chandler, Table of n, a(n) for n = 1..422
Dirk Frettlöh, "Tile Orientations with Distinct Frequencies", Ch. 1.5 in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity, 2017, see page 9.
Donald S. McDonald, World Mathematical Year 2000 Poster Competition
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A001983, A007511, A048610, A071383. Subsequence of A054994. Where records occur in A025441; corresponding number of ways is A060306.

c_old=-1;for(n=1,10000,c=0;for(i=1,floor(sqrt(n)),for(j=1,i-1,if(i^2+j^2==n,c+=1)));if(c>c_old,print1(n,", ");c_old=c)) \\ Derek Orr, Mar 15 2019

More terms from Randall L Rathbun, Jan 18 2002

Edited by Ray Chandler, Jan 12 2012

A001974 Numbers that are the sum of 3 distinct squares, i.e., numbers of the form x^2 + y^2 + z^2 with 0 <= x < y < z.

Original entry on oeis.org

5, 10, 13, 14, 17, 20, 21, 25, 26, 29, 30, 34, 35, 37, 38, 40, 41, 42, 45, 46, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 65, 66, 68, 69, 70, 73, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 93, 94, 97, 98, 100, 101, 104, 105, 106, 107, 109, 110, 113

Offset: 1

N. J. A. Sloane

Also: Numbers which are the sum of two or three distinct nonzero squares. - M. F. Hasler, Feb 03 2013

According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 distinct squares (i.e., is in A001974 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627, ?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017

			5 = 0^2 + 1^2 + 2^2.

T. D. Noe, Table of n, a(n) for n=1..1000
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
Index entries for sequences related to sums of squares

Cf. A001983, A024803, A025339, A025442, A004432.

Cf. A004436 (complement).

r[n_] := Reduce[0 <= x < y < z && x^2 + y^2 + z^2 == n, {x, y, z}, Integers]; ok[n_] := r[n] =!= False; Select[ Range[113], ok] (* Jean-François Alcover, Dec 05 2011 *)

from itertools import combinations
def aupto(lim):
  s = filter(lambda x: x <= lim, (i*i for i in range(int(lim**.5)+2)))
  s3 = set(filter(lambda x: x<=lim, (sum(c) for c in combinations(s, 3))))
  return sorted(s3)
print(aupto(113)) # Michael S. Branicky, May 10 2021

A001944 Numbers that are the sum of 4 distinct squares: of form w^2 + x^2 + y^2 + z^2 with 0 <= w < x < y < z.

Original entry on oeis.org

14, 21, 26, 29, 30, 35, 38, 39, 41, 42, 45, 46, 49, 50, 51, 53, 54, 56, 57, 59, 61, 62, 63, 65, 66, 69, 70, 71, 74, 75, 77, 78, 79, 81, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95, 98, 99, 101, 102, 104, 105, 106, 107, 109, 110, 111, 113, 114, 115, 116, 117

Offset: 1

N. J. A. Sloane

			14 = 0^2 + 1^2 + 2^2 + 3^2.

Cf. A001974, A001983, A001948, A001995.

nn = 20; Select[Union[Flatten[Table[a^2 + b^2 + c^2 + d^2, {a, 0, nn}, {b, a + 1, nn}, {c, b + 1, nn}, {d, c + 1, nn}]]], # <= nn^2 &] (* T. D. Noe, Aug 17 2012 *)

A004435 Positive integers that are not the sum of 2 distinct square integers.

Original entry on oeis.org

2, 3, 6, 7, 8, 11, 12, 14, 15, 18, 19, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 72, 75, 76, 77, 78, 79, 83, 84, 86, 87, 88, 91

Offset: 1

N. J. A. Sloane

nonn

A025435(a(n)) = 0. - Reinhard Zumkeller, Dec 20 2013

Cf. A001983 (complement).

a004435 n = a004435_list !! (n-1)
a004435_list = [x | x <- [1..], a025435 x == 0]
-- Reinhard Zumkeller, Dec 20 2013

Select[Range[100], Reduce[0 <= i < j && # == i^2 + j^2, {i, j}, Integers] === False &] (* Jean-François Alcover, Aug 01 2018 *)

A025302 Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way.

Original entry on oeis.org

5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 100, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 169, 173, 178, 180, 181, 193, 194, 197, 200, 202, 208, 212, 218, 225, 226, 229

Offset: 1

David W. Wilson

nonn

From Fermat's two squares theorem, every prime of the form 4k + 1 is a term (A002144). - Bernard Schott, Apr 15 2022

Donovan Johnson, Table of n, a(n) for n = 1..1000.
Art of Problem Solving, Fermat's Two Squares Theorem.
Index entries for sequences related to sums of squares.

Cf. A002144 (subsequence), A009000, A009003, A024507, A025441, A004431.

Cf. Subsequence of A001983; A004435.

a025302 n = a025302_list !! (n-1)
a025302_list = [x | x <- [1..], a025441 x == 1]

nn = 229; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i - 1}]; Flatten[Position[t, 1]] (* T. D. Noe, Apr 07 2011 *)
a[1] = 5; a[ n_] := a[n] = Module[ {s = a[n - 1], t = True, j}, While[ t, s++; Do[ If[ i^2 + (j = Floor[Sqrt[s - i^2]])^2 == s && i < j, t = False; Break], {i, Sqrt[s/2]}]]; s]; (* Michael Somos, Jan 20 2019 *)

from collections import Counter
from itertools import combinations
def aupto(lim):
  s = filter(lambda x: x <= lim, (i*i for i in range(1, int(lim**.5)+2)))
  s2 = filter(lambda x: x <= lim, (sum(c) for c in combinations(s, 2)))
  s2counts = Counter(s2)
  return sorted(k for k in s2counts if k <= lim and s2counts[k] == 1)
print(aupto(229)) # Michael S. Branicky, May 10 2021

A025441(a(n)) = 1. - Reinhard Zumkeller, Dec 20 2013

A132777 Nonsquare numbers which are the sum of 2 distinct squares.

Original entry on oeis.org

5, 10, 13, 17, 20, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 73, 74, 80, 82, 85, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 122, 125, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 164

Offset: 1

Rob Hubbard (hubbard.r(AT)gmail.com), Oct 08 2007

This is simply A001983 excluding the square numbers.

The squares of these numbers are also sums of two distinct nonzero squares. Including them gives A004431. - Jean-Christophe Hervé, Nov 11 2013

			5 = 4 + 1, and 5^2 = 16 + 9 ; 10 = 9 + 1, and 10^2 = 64 + 36. - _Jean-Christophe Hervé_, Nov 11 2013

Cf. A001983, A004431.

selQ[n_] := !IntegerQ[Sqrt[n]] && Select[ PowersRepresentations[n, 2, 2], 0 <= #[[1]] < #[[2]]&] != {};
Select[Range[1000], selQ] (* Jean-François Alcover, Apr 20 2023 *)

Definition corrected by Jean-Christophe Hervé, Nov 11 2013

A114090 Sums of 2 distinct nonnegative cubes.

Original entry on oeis.org

1, 8, 9, 27, 28, 35, 64, 65, 72, 91, 125, 126, 133, 152, 189, 216, 217, 224, 243, 280, 341, 343, 344, 351, 370, 407, 468, 512, 513, 520, 539, 559, 576, 637, 728, 729, 730, 737, 756, 793, 854, 855, 945, 1000, 1001, 1008, 1027, 1064, 1072, 1125, 1216, 1241, 1331

Offset: 1

Franklin T. Adams-Watters, Sep 14 2006

nonn

Except for zero, equivalently sums of at most 2 distinct positive cubes.

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

Cf. A024670, A003997, A004999, A000578, A001983.

N:= 2000: # to get all terms <= N
S:= select(`<=`,{seq(seq(a^3 + b^3, b = 0 .. a-1),a=0..floor(N^(1/3)))},N):
sort(convert(S,list)); # Robert Israel, May 31 2018

Select[Union@ Flatten[Table[i^3 + j^3, {i, 11}, {j, 0, i-1}]], # <= 11^3 &] (* Giovanni Resta, May 31 2018 *)
Module[{upto=15},Select[Union[Total/@Subsets[Range[0,upto]^3,{2}]],#<= upto^3&]] (* Harvey P. Dale, Sep 07 2019 *)

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

cp's OEIS Frontend

A003995 Sum of (any number of) distinct squares: of form r^2 + s^2 + t^2 + ... with 0 <= r < s < t < ...

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A025435 Number of partitions of n into 2 distinct squares.

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A052199 Numbers that are expressible as the sum of 2 distinct positive squares in more ways than any smaller number.

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A001974 Numbers that are the sum of 3 distinct squares, i.e., numbers of the form x^2 + y^2 + z^2 with 0 <= x < y < z.

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A001944 Numbers that are the sum of 4 distinct squares: of form w^2 + x^2 + y^2 + z^2 with 0 <= w < x < y < z.

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A004435 Positive integers that are not the sum of 2 distinct square integers.

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A025302 Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way.

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A294774 a(n) = 2n^2 + 2n + 5.