A001481 -id:A001481 - cp's OEIS frontend

A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 146, 149, 157, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 305, 313, 314, 317, 337, 346, 349

Offset: 1

Steven Finch

nonn

Primitively but not imprimitively represented by x^2 + y^2.
The disjoint union of {1}, A003654, and A031398. - Max Alekseyev, Mar 09 2010
Squarefree members of A202057. - Artur Jasinski, Dec 10 2011
Union of A231754 and 2*A231754. Squarefree numbers whose prime factors are in A002313. - Robert Israel, Aug 23 2017
It appears that a(n) is the n-th index, k, such that f(k) = 2, where f(k) = 3*(Sum_{i=1..k} floor(i^2/k)) - k^2 (see A175908). - John W. Layman, May 16 2011

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see page 123.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Index entries for sequences related to sums of squares

Cf. A001481, A008784, A022544, A034023, A175908.

Cf. also A000290, A010052, A005117, A084349, A002313, A231754.

a020893 n = a020893_list !! (n-1)
a020893_list = filter (\x -> any (== 1) $ map (a010052 . (x -)) $
                             takeWhile (<= x) a000290_list) a005117_list
-- Reinhard Zumkeller, May 28 2015

N:= 1000: # to get all terms <= N
R:= {1,2}:
p:= 2:
do
p:= nextprime(p);
if p > N then break fi;
if p mod 4 <> 1 then next fi;
R:= R union select(`<=`,map(`*`,R,p),N);
od:
sort(convert(R,list)); # Robert Israel, Aug 23 2017

lim = 17; t = Join[{1}, Select[Union[Flatten[Table[x^2 + y^2, {x, lim}, {y, x}]]], # < lim^2 && SquareFreeQ[#] &]]
Select[Union[Total/@Tuples[Range[0,20]^2,2]],SquareFreeQ] (* Harvey P. Dale, Jul 26 2017 *)
Block[{nn = 350, p}, p = {1, 2}~Join~Select[Prime@ Range@ PrimePi@ nn, Mod[#, 4] == 1 &]; Select[Range@ nn, And[SquareFreeQ@ #, SubsetQ[p, FactorInteger[#][[All, 1]]]] &]] (* Michael De Vlieger, Aug 23 2017 *)
(* or *)
Select[Range[350], SquareFreeQ@ # && ! MemberQ[Mod[First /@ FactorInteger@ #, 4], 3] &] (* Giovanni Resta, Aug 25 2017 *)

is(n)=my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1 || f[i,1]%4==3, return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015

from itertools import count, islice
from sympy import factorint
def A020893_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 and e == 1 for p, e in factorint(n).items()),count(1))
A020893_list = list(islice(A020893_gen(),30)) # Chai Wah Wu, Jun 28 2022

a(n) ~ k*n*sqrt(log n), where k = 2.1524249... = A013661/A064533. - Charles R Greathouse IV, Apr 20 2015

Edited by N. J. A. Sloane, Aug 30 2017

A057961 Number of points in square lattice covered by a disc centered at (0,0) as its radius increases.

Original entry on oeis.org

1, 5, 9, 13, 21, 25, 29, 37, 45, 49, 57, 61, 69, 81, 89, 97, 101, 109, 113, 121, 129, 137, 145, 149, 161, 169, 177, 185, 193, 197, 213, 221, 225, 233, 241, 249, 253, 261, 277, 285, 293, 301, 305, 317, 325, 333, 341, 349, 357, 365, 373, 377, 385, 401, 405, 421

Offset: 1

Ken Takusagawa, Oct 15 2000

Useful for rasterizing circles.

Conjecture: the number of lattice points in a quadrant of the disk is equal to A000592(n-1). - L. Edson Jeffery, Feb 10 2014

			a(2)=5 because (0,0); (0,1); (0,-1); (1,0); (-1,0) are covered by any disc of radius between 1 and sqrt(2).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.

T. D. Noe, Table of n, a(n) for n=1..1000
L. Edson Jeffery, Illustration of first few terms.

Cf. A004018, A004020, A005883, A057962. Distinct terms of A057655.

Cf. A000404, A001481, A232499.

max = 100; A001481 = Select[Range[0, 4*max], SquaresR[2, #] != 0 &]; Table[SquaresR[2, A001481[[n]]], {n, 1, max}] // Accumulate (* Jean-François Alcover, Oct 04 2013 *)

A336820 A(n,k) is the n-th number that is a sum of at most k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 3, 5, 5, 0, 1, 2, 3, 8, 8, 6, 0, 1, 2, 3, 4, 9, 9, 7, 0, 1, 2, 3, 4, 16, 10, 10, 8, 0, 1, 2, 3, 4, 5, 17, 16, 13, 9, 0, 1, 2, 3, 4, 5, 32, 18, 17, 16, 10, 0, 1, 2, 3, 4, 5, 6, 33, 19, 24, 17, 11, 0, 1, 2, 3, 4, 5, 6, 64, 34, 32, 27, 18, 12

Offset: 1

Alois P. Heinz, Aug 04 2020

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,   0,   0,   0,  0,  0, ...
   1,  1,  1,  1,  1,  1,   1,   1,   1,  1,  1, ...
   2,  2,  2,  2,  2,  2,   2,   2,   2,  2,  2, ...
   3,  4,  3,  3,  3,  3,   3,   3,   3,  3,  3, ...
   4,  5,  8,  4,  4,  4,   4,   4,   4,  4,  4, ...
   5,  8,  9, 16,  5,  5,   5,   5,   5,  5,  5, ...
   6,  9, 10, 17, 32,  6,   6,   6,   6,  6,  6, ...
   7, 10, 16, 18, 33, 64,   7,   7,   7,  7,  7, ...
   8, 13, 17, 19, 34, 65, 128,   8,   8,  8,  8, ...
   9, 16, 24, 32, 35, 66, 129, 256,   9,  9,  9, ...
  10, 17, 27, 33, 36, 67, 130, 257, 512, 10, 10, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-11 give: A001477(n-1), A001481, A004825, A004833, A004845, A004857, A004869, A004881, A004893, A004905, A004917.
A(n+j,n) for j=0-3 give: A001477(n-1), A000027, A000079, A000051.
Cf. A336725.

A:= proc() local l, w, A; l, w, A:= proc() [] end, proc() [] end,
      proc(n, k) option remember; local b; b:=
        proc(x, y) option remember; `if`(x<0 or y<1, {},
          {0, b(x, y-1)[], map(t-> t+l(k)[y], b(x-1, y))[]})
        end;
        while nops(w(k)) < n do forget(b);
          l(k):= [l(k)[], (nops(l(k))+1)^k];
          w(k):= sort([select(h-> h

b[n_, k_, i_, t_] := b[n, k, i, t] = n == 0 || i > 0 && t > 0 && (b[n, k, i - 1, t] || i^k <= n && b[n - i^k, k, i, t - 1]);
A[n_, k_] := A[n, k] = Module[{m}, For[m = 1 + If[n == 1, -1, A[n - 1, k]], !b[m, k, m^(1/k) // Floor, k], m++]; m];
Table[A[n, 1+d-n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, Dec 03 2020, using Alois P. Heinz's code for columns *)

A(n,k) = n-1 for n <= k+1.

A018786 Numbers that are the sum of two 4th powers in more than one way.

Original entry on oeis.org

635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752

Offset: 1

David W. Wilson

nonn

Since 4th powers are squares, this is a subsequence of A024508, the analog for squares. Sequence A001235 is the analog for third powers (taxicab numbers). Sequence A255351 lists max {a,b,c,d} where a^4 + b^4 = c^4 + d^4 = a(n), while A255352 lists the whole quadruples (a,b,c,d). - M. F. Hasler, Feb 21 2015

			a(1) = 59^4 + 158^4 = 133^4 + 134^4.
a(2) = 7^4 + 239^4 = 157^4 + 227^4. Note the remarkable coincidence that here all of {7, 239, 157, 227} are primes. The next larger solution with this property is 17472238301875630082 = 62047^4 + 40351^4 = 59693^4 + 46747^4. - _M. F. Hasler_, Feb 21 2015

R. K. Guy, Unsolved Problems in Number Theory, D1.

Mia Muessig, Table of n, a(n) for n = 1..30000 (terms 1..111 from Vincenzo Librandi, terms 112..4359 from Sean A. Irvine)
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
Mia Muessig, Julia code for finding general taxicab numbers
Eric Weisstein's World of Mathematics, Biquadratic Number.
Eric Weisstein's World of Mathematics, Diophantine Equation.

Subsequence of A003336 (and hence A004831) and A024508 (and hence A001481).

Cf. A001235, A003336, A003824, A309762.

Select[ Split[ Sort[ Flatten[ Table[x^4 + y^4, {x, 1, 1000}, {y, 1, x}]]]], Length[#] > 1 & ][[All, 1]] (* Jean-François Alcover, Jul 26 2011 *)

n=4;L=[];for(b=1,999,for(a=1,b,t=a^n+b^n;for(c=a+1,sqrtn(t\2,n),ispower(t-c^n,n)||next;print1(t",")))) \\ M. F. Hasler, Feb 21 2015

list(lim)=my(v=List()); for(a=134,sqrtnint(lim,4)-1, my(a4=a^4); for(b=sqrtnint((4*a^2 + 6*a + 4)*a,4)+1,min(sqrtnint(lim-a4,4),a), my(t=a4+b^4); for(c=a+1,sqrtnint(lim,4), if(ispower(t-c^4,4), listput(v,t); break)))); Set(v) \\ Charles R Greathouse IV, Jul 12 2024

A weak lower bound: a(n) >> n^2. - Charles R Greathouse IV, Jul 12 2024

A216451 Numbers which are simultaneously of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+7y^2, all with x>0, y>0.

Original entry on oeis.org

193, 337, 457, 673, 772, 1009, 1033, 1129, 1201, 1297, 1348, 1737, 1801, 1828, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2692, 2713, 2857, 3033, 3049, 3088, 3217, 3313, 3361, 3529, 3600, 3697, 3889, 4036, 4057, 4113, 4132, 4153, 4201, 4516, 4561, 4624, 4657

Offset: 1

V. Raman, Sep 07 2012

nonn

A number can be written as a^2+b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
A number can be written as a^2+2*b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
A number can be written as a^2+3*b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
A number can be written as a^2+7*b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power. Also the power of 2 should not be 1, if it can be written in the form a^2+7*b^2.

V. Raman, Table of n, a(n) for n = 1..1000

Cf. A154777, A092572.

Intersection of A001481, A002479, A003136 and A020670, omitting squares. See also A216500. - N. J. A. Sloane, Sep 11 2012

nn = 4657; lim = Floor[Sqrt[nn]]; t1 = Select[Union[Flatten[Table[a^2 + b^2, {a, lim}, {b, lim}]]], # <= nn &]; t2 = Select[Union[Flatten[Table[a^2 + 2*b^2, {a, lim}, {b, lim/Sqrt[2]}]]], # <= nn &]; t3 = Select[Union[Flatten[Table[a^2 + 3*b^2, {a, lim}, {b, lim/Sqrt[3]}]]], # <= nn &]; t7 = Select[Union[Flatten[Table[a^2 + 7*b^2, {a, lim}, {b, lim/Sqrt[7]}]]], # <= nn &]; Intersection[t1, t2, t3, t7] (* T. D. Noe, Sep 08 2012 *)

Definition clarified by N. J. A. Sloane, Sep 11 2012

A234300 Number of unit squares, aligned with a Cartesian grid, partially encircled along the edge of the first quadrant of a circle centered at the origin ordered by increasing radius.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 5, 3, 5, 4, 5, 5, 7, 5, 7, 5, 7, 7, 9, 7, 9, 8, 9, 7, 9, 7, 11, 9, 11, 9, 11, 10, 11, 9, 11, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 13, 15, 12, 15, 13, 15, 13, 15, 13, 15, 13, 15, 15, 17, 13, 17, 15, 17, 16, 17, 15, 17, 15, 17, 15, 17, 17, 19, 17, 19, 15, 19, 17, 19, 17, 19, 17, 19, 18, 19, 17, 21, 19, 21, 19, 21, 19, 21, 19, 21, 19, 21, 19, 21

Offset: 0

Rajan Murthy, Dec 22 2013

nonn

The first decrease from a(4) = 3 to a(5) = 2 occurs when the radius squared increases from an arbitrary position between 1 and 2 (when 3 squares are on the edge) to exactly 2 (when only 2 squares are on the edge because the circle of square radius 2 passes through the upper right corner on the y=x line). Similar decreases occur when the circle passes through other upper right corners. At least some (if not all) adjacent duplicates occur when the square radius corresponds to a perfect square, that is a corner which is only a lower right corner, i.e., on the y = 0 line. For example, a(6)=a(7)=3 occurs when, for n = 6 , a(n) corresponding to the interval between 2 and 4; and, for n=7, a(n) corresponding to the exact square radius of 4. Some of the confusion may come from the fact that for odd n, there is a unique circle corresponding to elements of a(n) (passing through the corner of specific square(s) on the grid), while for even n, there is a set of circles with a range of radii (which do not pass through corners) corresponding to the elements of a(n). It seems easier to organize the concept in terms of intervals and corners for the sake of consistency.

a(n) is even when the radius squared corresponds to an element of A024517.

			At radius 0, there are no partially filled squares.  At radius >1 but < sqrt(2), there are 3 partially filled squares along the edge of the circle.  At radius = sqrt(2), there are 2 partially filled squares along the edge of the circle.

Rajan Murthy, Table of n, a(n) for n = 0..4999

Cf. A001481 (corresponds to the square radius of alternate entries), A232499 (number of completely encircled squares when the radii are indexed by A000404), A235141 (first differences), A024517.

A237708 is the analog for the 3-dimensional Cartesian lattice and A239353 for the 4-dimensional Cartesian lattice.

function index = n_edgeindex (N)
    if N < 1 then
        N = 1
    end
    N = floor(N)
    i = 0:ceil(N/2)
    i = i^2
    index = i
    for j = 1:length(i)
       index = [index i+ i(j).*ones(i)]
    end
    index = unique(index)
    index = index(1:ceil(N))
    d = diff(index)/2
    d = d +  index(1:length(d))
    index = gsort([index d],"g","i")
    index = index(1:N)
endfunction
function l = n_edge_n (i)
        l=0
        h=0
        while (i > (2*h^2))
            h=h+1
        end
        if i < (2*h^2) then
                l = l+1
        end
        if i >1 then
            t=[0 1]
           while (i>max(t))
               t = [t (sqrt(max(t))+1)^2]
           end
        for j = 1:h
           b=t
           t=[2*(j)^2 (j+1)^2 + (j)^2]
           while (i>max(t))
               t = [t (sqrt(max(t)-(j)^2)+1)^2 + (j)^2]
           end
           l = l+ 2*(length(b)-length(t))
           if max(t) == i then
               l = l-2
           end
        end
       end
endfunction
function a =n_edge (N)
    if N <1 then
        N =1
    end
    N = floor(N)
    a= []
    index = n_edgeindex(N)
    for i = index
        a = [a n_edge_n(i)]
    end
endfunction

a(2k+1) = a(2k) + 2*A000161(A001481(k+1)) - A010052(A001481(k+1)/2). - Rajan Murthy, Jan 14 2013

a(2k) = a(2k-1) - 2*(A000161(A001481(k+1)) - A010052(A001481(k+1))) + A010052(A001481(k+1)/2). - Rajan Murthy, Jan 14 2013

A262311 Number of ordered ways to write n = x^2 + y^2 + phi(z^2) (0 <= x <= y and z > 0) with y or z prime, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 1, 1, 3, 3, 2, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 1, 2, 2, 3, 3, 2, 1, 3, 3, 1, 2, 2, 2, 3, 4, 4, 1, 3, 2, 6, 3, 2, 4, 4, 3, 1, 3, 4, 5, 5, 3, 3, 3, 4, 4, 8, 4, 3, 5, 4, 2, 2, 3, 6, 6, 1, 2, 5, 3, 2, 4, 5, 2, 2, 2, 3, 3, 2, 3, 6, 3, 2, 3, 3, 4, 4, 3, 3, 4, 5, 4, 3, 3, 1, 4, 3, 2, 3

Offset: 1

Zhi-Wei Sun, Oct 01 2015

nonn

Conjecture: a(n) > 0 for all n > 1.
I have verified this for n up to 10^6, and found that a(n) = 1 for the following 67 values of n: 2, 3, 4, 5, 8, 9, 13, 23, 29, 32, 39, 48, 68, 96, 108, 140, 144, 215, 264, 268, 324, 328, 384, 396, 404, 460, 471, 476, 500, 503, 684, 716, 759, 764, 768, 788, 860, 908, 936, 1032, 1076, 1112, 1148, 1164, 1259, 1344, 1399, 1443, 1484, 1503, 1551, 1839, 1868, 2088, 2723, 2883, 3744, 4296, 5963, 6804, 8328, 9680, 10331, 11948, 21524, 39716, 94415. It seems that a(n) = 1 for no other values of n.
It is easy to see that all the numbers phi(n^2) = n*phi(n) (n = 1,2,3,...) are pairwise distinct.
See also A262781 for a similar conjecture.

			a(2) = 1 since 2 = 0^2 + 0^2 + phi(2^2) with 2 prime.
a(5) = 1 since 5 = 0^2 + 2^2 + phi(1^2) with 2 prime.
a(23) = 1 since 23 = 1^2 + 4^2 + phi(3^2) with 3 prime.
a(29) = 1 since 29 = 0^2 + 3^2 + phi(5^2) with 3 and 5 both prime.
a(48) = 1 since 48 = 2^2 + 2^2 + phi(10^2) with 2 prime.
a(96) = 1 since 96 = 3^2 + 9^2 + phi(3^2) with 3 prime.
a(140) = 1 since 140 = 7^2 + 7^2 + phi(7^2) with 7 prime.
a(471) = 1 since 471 = 0^2 + 19^2 + phi(11^2) with 19 and 11 both prime.
a(476) = 1 since 476 = 8^2 + 16^2 + phi(13^2) with 13 prime.
a(936) = 1 since 936 = 4^2 + 30^2 + phi(5^2) with 5 prime.
a(1112) = 1 since 1112 = 23^2 + 23^2 + phi(9^2) with 23 prime.
a(1839) = 1 since 1839 = 3^2 + 30^2 + phi(31^2) with 31 prime.
a(1868) = 1 since 1868 = 2^2 + 2^2 + phi(62^2) with 2 prime.
a(2088) = 1 since 2088 = 15^2 + 39^2 + phi(19^2) with 19 prime.
a(2723) = 1 since 2723 = 34^2 + 35^2 + phi(19^2) with 19 prime.
a(2883) = 1 since 2883 = 21^2 + 44^2 + phi(23^2) with 23 prime.
a(3744) = 1 since 3744 = 4^2 + 54^2 + phi(29^2) with 29 prime.
a(4296) = 1 since 4296 = 26^2 + 60^2 + phi(5^2) with 5 prime.
a(5963) = 1 since 5963 = 26^2 + 59^2 + phi(43^2) with 59 and 43 both prime.
a(6804) = 1 since 6804 = 40^2 + 72^2 + phi(5^2) with 5 prime.
a(8328) = 1 since 8328 = 1^2 + 39^2 + phi(83^2) with 83 prime.
a(9680) = 1 since 9680 = 68^2 + 70^2 + phi(13^2) with 13 prime.
a(10331) = 1 since 10331 = 17^2 + 100^2 + phi(7^2) with 7 prime.
a(11948) = 1 since 11948 = 5^2 + 109^2 + phi(7^2) with 109 and 7 both prime.
a(21524) = 1 since 21524 = 59^2 + 109^2 + phi(79^2) with 109 and 79 both prime.
a(39716) = 1 since 39716 = 5^2 + 17^2 + phi(199^2) with 17 and 199 both prime.
a(94415) = 1 since 94415 = 115^2 + 178^2 + phi(223^2) with 223 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000290, A001481, A002618, A262747, A262781.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=EulerPhi[n^2]
Do[r=0;Do[If[f[z]>n,Goto[aa]];Do[If[SQ[n-f[z]-x^2]&&(PrimeQ[z]||PrimeQ[Sqrt[n-f[z]-x^2]]),r=r+1],{x,0,Sqrt[(n-f[z])/2]}];Label[aa];Continue,{z,1,n}];Print[n," ",r];Continue,{n,1,100}]

A069011 Triangle with T(n,k) = n^2 + k^2.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 18, 16, 17, 20, 25, 32, 25, 26, 29, 34, 41, 50, 36, 37, 40, 45, 52, 61, 72, 49, 50, 53, 58, 65, 74, 85, 98, 64, 65, 68, 73, 80, 89, 100, 113, 128, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200

Offset: 0

Henry Bottomley, Apr 02 2002

For any i,j >=0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013
A227481(n) = number of squares in row n. - Reinhard Zumkeller, Oct 11 2013
Norm of the complex numbers n +- i*k and k +- i*n, where i denotes the imaginary unit. - Stefano Spezia, Aug 07 2025

			Triangle T(n,k) begins:
    0;
    1,   2;
    4,   5,   8;
    9,  10,  13,  18;
   16,  17,  20,  25,  32;
   25,  26,  29,  34,  41,  50;
   36,  37,  40,  45,  52,  61,  72;
   49,  50,  53,  58,  65,  74,  85,  98;
   64,  65,  68,  73,  80,  89, 100, 113, 128;
   81,  82,  85,  90,  97, 106, 117, 130, 145, 162;
  100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.
Column k=0 gives A000290.
Main diagonal gives A001105.
Row sums give A132124.
T(2n,n) gives A033429.

a069011 n k = a069011_tabl !! n !! k
a069011_row n = a069011_tabl !! n
a069011_tabl = map snd $ iterate f (1, [0]) where
   f (i, xs@(x:_)) = (i + 2, (x + i) : zipWith (+) xs [i + 1, i + 3 ..])
-- Reinhard Zumkeller, Oct 11 2013

Table[n^2 + k^2, {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Aug 07 2025 *)

T(n+1,k+1) = T(n,k) + 2*(n+k+1), k=0..n; T(n+1,0) = T(n,0) + 2*n + 1. - Reinhard Zumkeller, Oct 11 2013

G.f.: x*(1 + 2*y + 5*x^3*y^2 - x^2*y*(2 + 5*y) + x*(1 - 4*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Aug 04 2025

A125022 Numbers with a unique partition as the sum of 2 squares x^2 + y^2.

Original entry on oeis.org

0, 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, 68, 72, 73, 74, 80, 81, 82, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 121, 122, 128, 136, 137, 144, 146, 148, 149, 153, 157, 160, 162, 164, 173, 178, 180, 181

Offset: 1

Artur Jasinski, Nov 16 2006

nonn

A000161(a(n)) = 1. [Reinhard Zumkeller, Aug 16 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002145, A124982, A125021, A034026.

Cf. A025284, A081324, A022544, A001481, A118882.

import Data.List (elemIndices)
a125022 n = a125022_list !! (n-1)
a125022_list = elemIndices 1 a000161_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range[0,200],Length@PowersRepresentations[#,2,2]==1&] (* Giorgos Kalogeropoulos, Mar 21 2021 *)

a(n) = A125021(n)/2.

Name edited by Giorgos Kalogeropoulos, Mar 21 2021

A140612 Integers k such that both k and k+1 are the sum of 2 squares.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 17, 25, 36, 40, 49, 52, 64, 72, 73, 80, 81, 89, 97, 100, 116, 121, 136, 144, 145, 148, 169, 180, 193, 196, 225, 232, 233, 241, 244, 256, 260, 288, 289, 292, 305, 313, 324, 337, 360, 361, 369, 388, 400, 404, 409, 424, 441, 449, 457

Offset: 1

Franklin T. Adams-Watters, May 19 2008

Equivalently, nonnegative k such that k*(k+1) is the sum of two squares.
Also, nonnegative k such that k*(k+1)/2 is the sum of two squares. This follows easily from the "sum of two squares theorem": x is the sum of two (nonnegative) squares iff its prime factorization does not contain p^e where p == 3 (mod 4) and e is odd. - Robert Israel, Mar 26 2018
Trivially, sequence includes all positive squares.

			40 = 6^2 + 2^2, 41 = 5^2 + 4^2, so 40 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A000404, A001481, A002378, A050795, A073613.

[k:k in [0..460]| forall{k+a: a in [0,1]|NormEquation(1, k+a) eq true}]; // Marius A. Burtea, Oct 08 2019

(*M6*) A1 = {}; Do[If[SquaresR[2, n (n + 1)/2] > 0, AppendTo[A1, n]], {n, 0, 1500}]; A1
Join[{0}, Flatten[Position[Accumulate[Range[500]], ?(SquaresR[2, #]> 0&)]]] (* _Harvey P. Dale, Jun 07 2015 *)
SequencePosition[Table[If[SquaresR[2,n]>0,1,0],{n,0,500}],{1,1}] [[All,1]]-1 (* Harvey P. Dale, Jul 28 2021 *)

from itertools import count, islice, starmap
from sympy import factorint
def A140612_gen(startvalue=0): # generator of terms >= startvalue
    for k in count(max(startvalue,0)):
        if all(starmap(lambda d, e: e % 2 == 0 or d % 4 != 3, factorint(k*(k+1)).items())):
            yield k
A140612_list = list(islice(A140612_gen(),20)) # Chai Wah Wu, Mar 07 2022

a(1)=0 prepended and edited by Max Alekseyev, Oct 08 2019

cp's OEIS Frontend

A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

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A057961 Number of points in square lattice covered by a disc centered at (0,0) as its radius increases.

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A336820 A(n,k) is the n-th number that is a sum of at most k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A018786 Numbers that are the sum of two 4th powers in more than one way.

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A216451 Numbers which are simultaneously of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+7y^2, all with x>0, y>0.

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A234300 Number of unit squares, aligned with a Cartesian grid, partially encircled along the edge of the first quadrant of a circle centered at the origin ordered by increasing radius.

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A262311 Number of ordered ways to write n = x^2 + y^2 + phi(z^2) (0 <= x <= y and z > 0) with y or z prime, where phi(.) is Euler's totient function given by A000010.

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