A084645 -id:A084645 - cp's OEIS frontend

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

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}.

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

A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

Original entry on oeis.org

7, 13, 14, 19, 21, 26, 28, 31, 35, 37, 38, 39, 42, 43, 52, 56, 57, 61, 62, 63, 65, 67, 70, 73, 74, 76, 77, 78, 79, 84, 86, 93, 95, 97, 103, 104, 105, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 134, 139, 140, 143, 146, 148, 151, 152, 154, 155, 156, 157, 158, 161

Offset: 1

Jean-Christophe Hervé, Nov 24 2013

nonn

Analog of A084645 for 120-degree angle triangles with integer sides.
Numbers with exactly one prime divisor of the form 6k+1 with multiplicity one.
Primitive elements of A050931.

			a(1) = 7 as 7^2 = 3^2 + 3*5 + 5^2.

Cf. A002476, A050931, A230780, A232436 (subsequence).

Cf. A084645, A232437, A248599, A254063, A254064.

r[k_] := Reduce[x>0 && y>0 && k^2 == x^2 + x y + y^2, {x, y}, Integers];
selQ[k_] := Which[rk = r[k]; rk === False, False, rk[[0]] === And && Length[rk] == 2, False, rk[[0]] === Or && Length[rk] == 2, True, True, False];
Select[Range[1000], selQ] (* Jean-François Alcover, Feb 20 2020 *)

Terms are obtained by the products A230780(k)*A002476(p) for k, p > 0, ordered by increasing values.

A254064 Positive integers whose square is expressible in exactly one way as x^2 + 6xy + y^2, with 0 < x < y.

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 102, 103, 105, 112, 113, 115, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146, 151, 153, 154, 155, 158, 164, 167

Offset: 1

Colin Barker, Jan 24 2015

nonn

Equivalently positive integers whose square is expressible in exactly one way as -x^2 + 2xy + y^2 with 0 < x < y by replacing (x,y) with (2x,x+y). As such this sequence represents the sum of legs that are unique to a single Pythagorean triangle. - Ray Chandler, Feb 18 2020

n is in the sequence iff A331671(n)=1. - Ray Chandler, Feb 26 2020

			7 is in the sequence because the only solution to x^2 + 6xy + y^2 = 49 with 0 < x < y is (x,y) = (2,3).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 750 terms from Colin Barker)

Cf. A084645, A232437, A248599, A254063, A331671.

s[n_] := Solve[0 < x < y && n^2 == x^2 + 6 x y + y^2, {x, y}, Integers];
Reap[For[n = 1, n < 200, n++, If[Length[s[n]]==1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 13 2020 *)

A248599 Positive integers whose square is expressible in exactly one way as x^2 + 4xy + y^2, with 0 < x < y.

Original entry on oeis.org

11, 13, 22, 23, 26, 33, 37, 39, 44, 46, 47, 52, 55, 59, 61, 65, 66, 69, 71, 73, 74, 77, 78, 83, 88, 91, 92, 94, 97, 99, 104, 107, 109, 110, 111, 115, 117, 118, 122, 130, 131, 132, 138, 141, 142, 146, 148, 154, 156, 157, 161, 165, 166, 167, 176, 177, 179, 181

Offset: 1

Colin Barker, Jan 21 2015

nonn

			11 is in the sequence because the only solution to x^2 + 4xy + y^2 = 121 with 0 < x < y is (x,y) = (4,5).

Colin Barker, Table of n, a(n) for n = 1..750

Cf. A232437, A084645, A254063, A254064.

A254063 Positive integers whose square is expressible in exactly one way as x^2 + 5xy + y^2, with 0 < x < y.

Original entry on oeis.org

5, 10, 15, 17, 20, 30, 34, 35, 37, 40, 41, 43, 45, 47, 51, 55, 59, 60, 65, 67, 68, 70, 74, 79, 80, 82, 83, 86, 89, 90, 94, 95, 101, 102, 105, 109, 110, 111, 115, 118, 119, 120, 123, 127, 129, 130, 131, 134, 135, 136, 140, 141, 145, 148, 151, 153, 155, 158

Offset: 1

Colin Barker, Jan 24 2015

nonn

			5 is in the sequence because the only solution to x^2 + 5xy + y^2 = 25 with 0 < x < y is (x,y) = (1,3).

Colin Barker, Table of n, a(n) for n = 1..750

Cf. A084645, A232437, A248599, A254064.

A232436 Numbers which are uniquely decomposable into x^2+xy+y^2, the unique decomposition being with two distinct nonzero x and y.

Original entry on oeis.org

7, 13, 19, 21, 28, 31, 37, 39, 43, 52, 57, 61, 63, 67, 73, 76, 79, 84, 93, 97, 103, 109, 111, 112, 117, 124, 127, 129, 139, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 193, 199, 201, 208, 211, 219, 223, 228, 229, 237, 241, 244, 252, 268

Offset: 1

Jean-Christophe Hervé, Nov 23 2013

nonn

These are the primitive elements of A024606, the integers which are expressible as x^2 + xy + y^2 with distinct nonzero x and y.
As a subsequence of A003136 (Loeschian numbers), the sequence is related with the triangular lattice: circles with radius sqrt(a(n)) centered at a grid point in this lattice hit exactly 12 points, cf. A004016.
Numbers with exactly one prime factor of form 6k+1 with multiplicity one and no prime factor of form 3k+2 with odd multiplicity, that is a(n) is of form 3^a*p*q^2, with a>=0, p a prime of form 6k+1, and q an integer with all its prime factors of form 3k+2. There is thus no square in the sequence.
From a(n) = 3^a*p*q^2, it is easily seen that sigma(a(n)) = 2 mod 6,
thus this sequence is a subsequence of A074628: the two sequences are equal up to a(308) = 1723; then A074628(309)= 1729 = a(1)*a(2)*a(3),  the famous Ramanujan's taxi number, and a(309) = A074628(310) = 1731.
The square of these numbers is also uniquely decomposable into the form x^2 + xy + y^2 with x and y > 0, thus this sequence is a subsequence of A232437.

			a(1)= 7 = 2^2+2+1, a(2)= 13 = 3^2+3+1. However 3 = 1+1+1 and 4 = 2^2+0*2+0 are not in the sequence because the unique decomposition of these numbers is not with two distinct nonzero numbers; 49, 147 are also excluded because there are two decompositions of these numbers (including one with equal or zero components x and y).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 6364 terms from Jean-Christophe Hervé)
A. Mazel, I. Stuhl, Y. Suhov, Hard-core configurations on a triangular lattice and Eisenstein primes, arXiv:1803.04041 [math.PR], 2018.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003136, A024606, A002476, A232437.
Cf. A004016, A034020 (A004016 = 0), A230781 (A004016 = 6).
Cf. (Analog for the square lattice) A230779, A001481, A004431, A002144, A004018, A084645.

r[k_] := Reduce[x != 0 && y != 0 && x != y && k == x^2 + x y + y^2, {x, y}, Integers];
selQ[k_] := If[IntegerQ[Sqrt[k]], False, Which[rk = r[k]; rk === False, False, rk[[0]] === And && Length[rk] == 2, True, rk[[0]] === Or && Length[rk] == 12, True, True, False]];
Select[Range[1000], selQ] (* Jean-François Alcover, Feb 20 2020 *)

Terms are obtained by the products A230781(k)*A002476(p) for k, p > 0, ordered by increasing values.

A004016(a(n))=12.

A232438 Squares or double-squares that are the sum of two distinct nonzero squares in exactly one way.

Original entry on oeis.org

25, 50, 100, 169, 200, 225, 289, 338, 400, 450, 578, 676, 800, 841, 900, 1156, 1225, 1352, 1369, 1521, 1600, 1681, 1682, 1800, 2025, 2312, 2450, 2601, 2704, 2738, 2809, 3025, 3042, 3200, 3362, 3364, 3600, 3721, 4050, 4624, 4900, 5202, 5329, 5408, 5476

Offset: 1

Jean-Christophe Hervé, Dec 01 2013

nonn

Subsequence of A004431 and A001481.

Numbers with exactly one prime factor of form 4k+1 with multiplicity 2, and without prime factor of form 4k+3 to an odd multiplicity.

			25 = 5^2 = 16+9; 50 = 2*5^2 = 49+1.

Jean-Christophe Hervé and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 368 terms from Jean-Christophe Hervé)

Cf. A001481, A004431, A004018, A125853 (A004018 = 4), A230779 (A004018 = 8), A025303 (A004018 = 16).

Analogs for square decompositions: A084645, A084646, A084647, A084648, A084649.

Select[Range[10^4], (IntegerQ[Sqrt[#]] || IntegerQ[Sqrt[#/2]]) && Count[ PowersRepresentations[#, 2, 2], {x_, y_} /; Unequal[0, x, y]] == 1 &]
(* or *) Select[Range[10^4], SquaresR[2, #] == 12 &] (* Jean-François Alcover, Dec 03 2013 *)

A004018(a(n)) = 12.

Terms are obtained by the products A125853(k)*A002144(p)^2 for k, p > 0, ordered by increasing values.

A350724 Hypotenuses for which there exists a unique integer-sided triangle and where the other two sides of this triangle are also hypotenuses for which there exists a unique integer-sided triangle.

Original entry on oeis.org

89, 109, 149, 173, 178, 218, 233, 267, 293, 298, 327, 346, 349, 356, 409, 421, 436, 447, 449, 461, 466, 509, 519, 534, 586, 596, 601, 613, 623, 653, 654, 692, 698, 699, 709, 712, 761, 763, 801, 818, 842, 872, 877, 879, 894

Offset: 1

Robin Jehn, Feb 03 2022

nonn

			89^2 = 39^2 + 80^2. This is the only integer decomposition. Also 39^2 = 15^2 + 36^2 and 80^2 = 48^2 + 64^2 are unique decompositions.

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

Cf. A046080, A084645.

A379985 Numbers k such that k^2 is of the form b^2 + (4c)^2 where bc is squarefree.

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 61, 65, 85, 109, 137, 145, 149, 157, 169, 173, 193, 197, 205, 221, 229, 241, 265, 269, 293, 305, 325, 365, 377, 401, 409, 421, 433, 445, 485, 505, 533, 541, 557, 565, 569, 629, 673, 685, 689, 701, 709, 725, 761, 773, 797

Offset: 1

Lei Zhou, Jan 07 2025

nonn

It is known that the sum of squares of two odd numbers cannot be a square number, and when the sum of square of two numbers is the square of an odd number, the even one among the two numbers has to be multiple of 4. Thus the Mathematica program will not miss any entries.
a(n) == 1 (mod 4).
Numbers 4x^2 + y^2 where x, y are coprime numbers such that y is odd and x, y, 2x+y, 2x-y are squarefree. - Yifan Xie, Jan 09 2025, corrected by Robert Israel, Feb 03 2025

			5 is a term since 5^2 = 3^2 + (4*1)^2 and 3*1 is squarefree.
149 is a term since 149^2 = 51^2 + (4*35)^2 and 51*35 = 3*5*7*17 is squarefree.

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

Subsequence of A009003.

Cf. A005117, A084645, A009000, A057100.

N:= 1000: # for terms <= N
Res:= {}:
for x from 1 while 4*x^2 < N do
  if not numtheory:-issqrfree(x) then next fi;
  for y from 1 by 2  while 4*x^2 + y^2 <= N do
    if igcd(x,y) = 1 and andmap(numtheory:-issqrfree,[y,2*x+y,2*x-y]) then Res:= Res union  {4*x^2 + y^2} fi
od od:
sort(convert(Res,list)); # Robert Israel, Feb 03 2025

a = {}; Do[m = n^2; b = n; While[b = b - 2; b > 1, k = m - b^2; If[c = Sqrt[k]/4; IntegerQ[c] && SquareFreeQ[b*c], AppendTo[a, n]]], {n, 5, 800, 2}]; a

Edited by Robert Israel, Feb 03 2025

cp's OEIS Frontend

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

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A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

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A254064 Positive integers whose square is expressible in exactly one way as x^2 + 6xy + y^2, with 0 < x < y.

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A248599 Positive integers whose square is expressible in exactly one way as x^2 + 4xy + y^2, with 0 < x < y.

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A254063 Positive integers whose square is expressible in exactly one way as x^2 + 5xy + y^2, with 0 < x < y.

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A232436 Numbers which are uniquely decomposable into x^2+xy+y^2, the unique decomposition being with two distinct nonzero x and y.

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A232438 Squares or double-squares that are the sum of two distinct nonzero squares in exactly one way.

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A379985 Numbers k such that k^2 is of the form b^2 + (4c)^2 where bc is squarefree.