A004431 -id:A004431 - cp's OEIS frontend

A224981 Numbers that are the sum of exactly 6 distinct nonzero squares.

91, 104, 115, 119, 124, 130, 131, 136, 139, 143, 146, 147, 151, 152, 154, 155, 156, 159, 160, 163, 164, 166, 167, 168, 169, 170, 171, 175, 176, 178, 179, 180, 181, 182, 184, 187, 188, 190, 191, 192, 194, 195, 196, 199, 200, 201, 202, 203, 204, 206, 207, 208

Offset: 1

Reinhard Zumkeller, Apr 22 2013

nonn

			a(1) = 1 + 4 + 9 + 16 + 25 + 36 = 91 = A000330(6);
a(2) = 1 + 4 + 9 + 16 + 25 + 49 = 104;
a(3) = 1 + 4 + 9 + 16 + 36 + 49 = 115;
a(4) = 1 + 4 + 9 + 16 + 25 + 64 = 119;
a(5) = 1 + 4 + 9 + 25 + 36 + 49 = 124.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Index entries for sequences related to sums of squares

Cf. A003995, A004431, A004432, A004433, A004434, A224982, A224983, A000290.

a224981 n = a224981_list !! (n-1)
a224981_list = filter (p 6 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)

nmax = 1000;
S[n_] := S[n] = Union[Total /@ Subsets[
     Range[Floor[Sqrt[n]]]^2, {6}]][[1 ;; nmax]];
S[nmax];
S[n = nmax + 1];
While[S[n] != S[n - 1], n++];
S[n] (* Jean-François Alcover, Nov 20 2021 *)

A224982 Numbers that are the sum of exactly 7 distinct nonzero squares.

Original entry on oeis.org

140, 155, 168, 172, 179, 185, 188, 191, 195, 196, 200, 203, 204, 205, 211, 212, 215, 217, 219, 220, 224, 225, 227, 230, 231, 232, 233, 235, 236, 239, 240, 243, 244, 245, 246, 247, 248, 251, 252, 254, 256, 257, 259, 260, 263, 264, 265, 267, 268, 269, 270, 271

Offset: 1

Reinhard Zumkeller, Apr 22 2013

nonn

			a(1) = 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140 = A000330(7);
a(2) = 1 + 4 + 9 + 16 + 25 + 36 + 64 = 155;
a(3) = 1 + 4 + 9 + 16 + 25 + 49 + 64 = 168;
a(4) = 1 + 4 + 9 + 16 + 25 + 36 + 81 = 172;
a(5) = 1 + 4 + 9 + 16 + 36 + 49 + 64 = 179.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Index entries for sequences related to sums of squares

Cf. A003995, A004431, A004432, A004433, A004434, A224981, A224983, A000290.

a224982 n = a224982_list !! (n-1)
a224982_list = filter (p 7 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)

nmax = 1000;
S[n_] := S[n] = Union[Total /@ Subsets[
     Range[Floor[Sqrt[n]]]^2, {7}]][[1 ;; nmax]];
S[nmax];
S[n = nmax + 1];
While[S[n] != S[n - 1], n++];
S[n] (* Jean-François Alcover, Nov 20 2021 *)

A224983 Numbers that are the sum of exactly 8 distinct nonzero squares.

Original entry on oeis.org

204, 221, 236, 240, 249, 255, 260, 261, 268, 269, 272, 276, 279, 281, 284, 285, 288, 289, 293, 295, 296, 299, 300, 303, 305, 306, 309, 311, 312, 316, 317, 320, 321, 323, 324, 325, 326, 327, 329, 332, 333, 335, 336, 337, 338, 339, 340, 341, 344, 345, 347, 348

Offset: 1

Reinhard Zumkeller, Apr 22 2013

nonn

			a(1) = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204 = A000330(8);
a(2) = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 81 = 221;
a(3) = 1 + 4 + 9 + 16 + 25 + 36 + 64 + 81 = 236;
a(4) = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 100 = 240;
a(5) = 1 + 4 + 9 + 16 + 25 + 49 + 64 + 81 = 249.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Index entries for sequences related to sums of squares

Cf. A003995, A004431, A004432, A004433, A004434, A224981, A224982, A000290.

a224983 n = a224983_list !! (n-1)
a224983_list = filter (p 8 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)

nmax = 1000;
S[n_] := S[n] = Union[Total /@ Subsets[
     Range[Floor[Sqrt[n]]]^2, {8}]][[1 ;; nmax]];
S[nmax];
S[n = nmax + 1];
While[S[n] != S[n - 1], n++];
S[n] (* Jean-François Alcover, Nov 20 2021 *)

A230779 Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.

Original entry on oeis.org

5, 10, 13, 17, 20, 26, 29, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212, 218, 226, 229, 232, 233, 234, 241, 244, 245, 257, 261, 269, 272

Offset: 1

Jean-Christophe Hervé, Nov 16 2013

nonn

Numbers with exactly one prime factor of form 4*k+1, that must have multiplicity one, and no prime factor of the form 4*k+3 with odd multiplicity. There is thus no square in the sequence.
These are the primitive elements of A004431, the integers which are the sum of two nonzero distinct squares.
Numbers such that A004018(a(n)) = 8.
The square of these numbers is also uniquely decomposable into a sum of two squares, thus this sequence is a subsequence of A084645.
Also a subsequence of A191217: the two sequences are equal up to a(76) = 320, then A191217(77) = 325, the value which is missing from this sequence, as a(77) = 328 = A191217(78). (3125 is also missing from this sequence, although present in A191217, and it is the 31st such number). - Corrected by Antti Karttunen, May 14 2022.
Numbers n such that n^3 is the sum of two nonzero squares in exactly two ways. - Altug Alkan, Jul 01 2016
Sequence A125022 (numbers with a unique partition as the sum of 2 squares x^2 + y^2), but without any terms of A028982 (squares and twice squares) that might occur there. - Antti Karttunen, May 14 2022

			a(1) = 5 = 4+1, a(2) = 10 = 9+1, a(3) =  13 = 9+4. However 2 = 1+1, 4 = 4+0, 8 = 4+4 are excluded because the unique decomposition of these numbers in two squares is not with two distinct nonzero squares; 25, 50, 100 are also excluded because there are two decompositions of these numbers in two squares (including one with equal or zero squares).

Antti Karttunen, Table of n, a(n) for n = 1..20000 (extending the previous b-file from Jean-Christophe Hervé, which contained terms up to the 1647th term 10009, but accidentally missed terms 8992 and 9376)
Eric Weisstein's World of Mathematics, Square Number
Wikipedia (fr), Théorème des deux carrés
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A001481, A004431, A002144, A028982, A353813 (characteristic function).
Subsequence of A004431, of A084645, of A125022, and of A191217.
Cf. A004018, A022554 (A004018 = 0), A125853 (A004018 = 4).

isok(n) = {f = factor(n); nb1 = 0; for (i=1, #f~, p = f[i, 1]; ep = f[i, 2]; if (p % 4 == 1, nb1 ++; if (ep != 1, return (0))); if (p % 4 == 3, if (ep % 2, return (0)));); return (nb1 == 1);} \\ Michel Marcus, Nov 17 2013

Terms are obtained by the products A125853(k)*A002144(p) for k, p > 0, ordered by increasing values.

{k | A004018(k) = 8}.

A286364 Compound filter: a(n) = P(A286361(n), A286363(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 1, 7, 3, 2, 2, 3, 2, 5, 1, 3, 7, 2, 3, 16, 2, 2, 2, 10, 3, 29, 2, 3, 5, 2, 1, 16, 3, 5, 7, 3, 2, 5, 3, 3, 16, 2, 2, 12, 2, 2, 2, 7, 10, 5, 3, 3, 29, 5, 2, 16, 3, 2, 5, 3, 2, 67, 1, 21, 16, 2, 3, 16, 5, 2, 7, 3, 3, 14, 2, 16, 5, 2, 3, 121, 3, 2, 16, 21, 2, 5, 2, 3, 12, 5, 2, 16, 2, 5, 2, 3, 7, 67, 10, 3, 5, 2, 3, 23, 3, 2, 29, 3, 5, 5, 2, 3

Offset: 1

Antti Karttunen, May 08 2017

nonn

This sequence packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027. These two components essentially give the prime signature of 4k+1 part and the prime signature of 4k+3 part, and they can be accessed from a(n) with functions A002260 and A004736. For example, A004431 lists all such numbers that the first component is larger than one and the second component is a perfect square.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A002260, A004431, A004736, A038722, A267099, A286361, A286363, A286365.

from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3))) # Indranil Ghosh, May 09 2017

(define (A286364 n) (* (/ 1 2) (+ (expt (+ (A286361 n) (A286363 n)) 2) (- (A286361 n)) (- (* 3 (A286363 n))) 2)))

a(n) = (1/2)*(2+((A286361(n)+A286363(n))^2) - A286361(n) - 3*A286363(n)).
Other identities. For all n >= 1:
a(A267099(n)) = A038722(a(n)).

A025303 Numbers that are the sum of 2 distinct nonzero squares in exactly 2 ways.

Original entry on oeis.org

65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 340, 365, 370, 377, 410, 442, 445, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 580, 585, 610, 625, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 820, 865, 884, 890, 901, 905, 949

Offset: 1

David W. Wilson

nonn

Numbers with exactly 2 distinct prime divisors of the form 4k+1, each with multiplicity one, or with only one prime divisor of this form with multiplicity 3, and with no prime divisor of the form 4k+3 to an odd multiplicity. - Jean-Christophe Hervé, Dec 01 2013

			65 = 5*13 = 64+1 = 49 + 16; 85 = 5*17 = 81+4 = 49+16; 125 = 5^3 = 121+4 = 100+25; 130 = 2*5*13 = 121+9 = 81+49. - _Jean-Christophe Hervé_, Dec 01 2013

Donovan Johnson, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares

Cf. A001481, A004431, A004018, A230779 (one way).

Cf. analogs for square decompositions: A084645, A084646, A084647, A084648, A084649.

nn = 949; 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, 2]] (* T. D. Noe, Apr 07 2011 *)

A004018(a(n)) = 16. - Jean-Christophe Hervé, Dec 01 2013

A155468 Numbers that are sums of 8th powers of 2 distinct positive integers.

Original entry on oeis.org

257, 6562, 6817, 65537, 65792, 72097, 390626, 390881, 397186, 456161, 1679617, 1679872, 1686177, 1745152, 2070241, 5764802, 5765057, 5771362, 5830337, 6155426, 7444417, 16777217, 16777472, 16783777, 16842752, 17167841, 18456832, 22542017, 43046722, 43046977, 43053282

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

nonn

			1^8 + 2^8 = 257, 1^8 + 3^8 = 6562, 2^8 + 3^8 = 6817, ...

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A003380, A088719 (distinct 7th), A088677 (distinct 6th), A088703, A088687, A024670 (distinct 3rd), A004431 (distinct 2nd).

lst={};e=8;Do[Do[x=a^e;Do[y=b^e;If[x+y==n,Print[n,",",Date[]];AppendTo[lst,n]],{b,Floor[(n-x)^(1/e)],a+1,-1}],{a,Floor[n^(1/e)],1,-1}],{n,4*8!}];lst

list(lim)=my(v=List(),t); lim\=1; for(m=2,sqrtnint(lim-1,8), t=m^8; for(n=1,min(sqrtnint(lim-t,8),m-1), listput(v,t+n^8))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017

8 more terms. - R. J. Mathar, Sep 07 2017

More terms from Chai Wah Wu, Nov 05 2017

A292313 Numbers that are the sum of three squares in arithmetic progression.

Original entry on oeis.org

75, 300, 507, 675, 867, 1200, 1875, 2028, 2523, 2700, 3468, 3675, 4107, 4563, 4800, 5043, 6075, 7500, 7803, 8112, 8427, 9075, 10092, 10800, 11163, 12675, 13872, 14700, 15987, 16428, 16875, 18252, 19200, 20172, 21675, 22707, 23763, 24300, 24843, 27075, 28227, 30000, 30603

Offset: 1

Antonio Roldán, Sep 14 2017

nonn

			75 = 1^2 + 5^2 + 7^2 = 1 + 25 + 49, with 25 - 1 = 49 - 25 = 24.
675 = 3^2 + 15^2 + 21^2 = 9 + 225 + 441, with 225 - 9 = 441 - 225 = 216.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subsequence of A033428.

Cf. A000290, A000378, A004431, A134422, A292309, A292310, A292314, A292316.

t=4; k=3; while(t<=13000, i=k; e=0; v=t+i; while(i>1&&e==0, if(issquare(v), m=3*t; e=1; print1(m,", ")); i+=-2; v+=i); k+=2; t+=k)

Sequence is 3*(distinct elements in A198385).

Numbers of the form 3*m^2 where 2*m^2 is in A004431. - Chai Wah Wu, Oct 05 2017

A335036 Smallest side c of the primitive triples (c,a,b) for integer triangles that have two perpendicular medians, ordered by increasing perimeter.

Original entry on oeis.org

13, 17, 25, 37, 41, 53, 61, 65, 85, 89, 101, 109, 113, 145, 145, 149, 157, 173, 181, 185, 193, 197, 221, 229, 233, 241, 257, 265, 269, 281, 277, 289, 293, 313, 317, 337, 349, 365, 365, 377, 377, 389, 397, 401, 409, 421, 433, 445, 461

Offset: 1

Bernard Schott, May 28 2020

nonn

If medians drawn from A and B are perpendicular in centroid G, then a^2 + b^2 = 5 * c^2, hence c is always the smallest odd side (see link Maths Challenge).
c = u^2 + v^2 for some u and v (see formula), so this sequence is subsequence of A004431.
For the corresponding primitive triples and miscellaneous properties, see A335034.
The repetitions for 145, 365, 377,... correspond to smallest sides for triangles with distinct perimeters (see examples).
This sequence is not increasing a(30) = 281 for triangle with perimeter = 1134 and a(31) = 277 for triangle with perimeter = 1148. The smallest side is not an increasing function of the perimeter of these triangles.

			The triples (145, 178, 271) and (145, 191, 262) correspond to triangles with respective perimeters equal to 594 and 598, so a(14) = a(15) = 145.
The triples (365, 418, 701) and (365, 509, 638) correspond to triangles with respective perimeters equal to 1484 and 1512, so a(38) = a(39) = 365.

Maths Challenge, Perpendicular medians, Problem with picture.

Subsequence of A004431.

Cf. A335034 (primitive triples), A335035 (corresponding perimeters), A335347 (middle side), A335348 (largest side), A335273 (even side).

mycmp(x, y) = {my(xp = vecsum(x), yp = vecsum(y)); if (xp!=yp, return (xp-yp)); return (x[1] - y[1]); }
lista(nn) = {my(vm = List(), vt, w); for (u=1, nn, for (v=1, nn, if (gcd(u, v) == 1, vt = 0; if ((u/v > 3) && ((u-3*v) % 5), vt = [2*(u^2-u*v-v^2), u^2+4*u*v-v^2, u^2+v^2]); if ((u/v > 1) && (u/v < 2) && ((u-2*v) % 5), vt = [2*(u^2+u*v-v^2), -u^2+4*u*v+v^2, u^2+v^2]); if (gcd(vt) == 1, listput(vm, vt));););); w = vecsort(apply(vecsort, Vec(vm)); , mycmp); vector(#w, k, w[k][1]);} \\ Michel Marcus, May 28 2020

a(n) = A335034(3n-2).
a(n) = A335035(n) - A335347(n) - A335348(n).
There exist two disjoint classes of such triangles, obtained with two distinct families of formulas: let u > v > 0 , u and v with different parities, gcd(u,v) = 1; if c is the smallest odd side, then:
1st class of triangles: c = u^2+v^2 with u/v > 3 and 5 doesn't divide u-3v,
2nd class of triangles: c = u^2+v^2 with 1 < u/v < 2 and 5 doesn't divide u-2v.

A125853 Squared radii of circles centered at a grid point in a square lattice hitting exactly 4 points. Indices k such that A004018(k)=4.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 49, 64, 72, 81, 98, 121, 128, 144, 162, 196, 242, 256, 288, 324, 361, 392, 441, 484, 512, 529, 576, 648, 722, 729, 784, 882, 961, 968, 1024, 1058, 1089, 1152, 1296, 1444, 1458, 1568, 1764, 1849, 1922, 1936, 2048, 2116, 2178, 2209

Offset: 1

Hugo Pfoertner, Jan 07 2007

nonn

From Jean-Christophe Hervé, Nov 17 2013: (Start)
Squares or double of squares that are not sum of two distinct nonzero squares.
Numbers without prime factor of form 4*k+1 and without prime factor of form 4*k+3 to an odd multiplicity.
The sequence is closed under multiplication. Primitive elements are 1, 2 and square of primes of form 4*k+3, that is union of {1, 2} and A087691.
Sequence A001481 (sum of two squares) is the union of {0}, this sequence and A004431 (sum of two distinct nonzero squares). These 4 sequences are all closed under multiplication. (End)

Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 18 2007, Table of n, a(n) for n = 1..501
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A004018, A001481, A004431.

for(n=1,100000,fctrs=factor(n);c=1;for(i=1,matsize(fctrs)[1],p4=fctrs[i,1]%4;if(p4==1 || (p4==3 && fctrs[i,2]%2==1), c=0)); if(c,print1(n","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 17 2007

Numbers of the form 2^e0 * 3^(2*e1) * 7^(2*e2) * 11^(2*e3) * ... * qk^(2*ek) where qk is the k-th prime of the form 4*n+3 (A002145). - Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 17 2007

cp's OEIS Frontend

A224981 Numbers that are the sum of exactly 6 distinct nonzero squares.

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A224982 Numbers that are the sum of exactly 7 distinct nonzero squares.

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A224983 Numbers that are the sum of exactly 8 distinct nonzero squares.

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A230779 Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.

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A286364 Compound filter: a(n) = P(A286361(n), A286363(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A025303 Numbers that are the sum of 2 distinct nonzero squares in exactly 2 ways.

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Mathematica

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A155468 Numbers that are sums of 8th powers of 2 distinct positive integers.

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Extensions

A292313 Numbers that are the sum of three squares in arithmetic progression.

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