A002479 -id:A002479 - cp's OEIS frontend

A155565 Intersection of A001481 and A020669: N = a^2 + b^2 = c^2 + 5d^2 for some integers a,b,c,d.

0, 1, 4, 5, 9, 16, 20, 25, 29, 36, 41, 45, 49, 61, 64, 80, 81, 89, 100, 101, 109, 116, 121, 125, 144, 145, 149, 164, 169, 180, 181, 196, 205, 225, 229, 241, 244, 245, 256, 261, 269, 281, 289, 305, 320, 324, 349, 356, 361, 369, 389, 400, 401, 404, 405, 409, 421

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155575 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155565(n,/* use optional 2nd arg to get other analogous sequences */c=[5,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,500, isA155565(n) & print1(n","))

A155566 Intersection of A001481 and A002481: N = a^2 + b^2 = c^2 + 6d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 25, 36, 40, 49, 58, 64, 73, 81, 90, 97, 100, 106, 121, 144, 145, 160, 169, 193, 196, 202, 225, 232, 241, 250, 256, 265, 289, 292, 298, 313, 324, 337, 346, 360, 361, 388, 394, 400, 409, 424, 433, 441, 457, 484, 490, 505, 522, 529, 538, 576

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155576 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155566(n,/* use optional 2nd arg to get other analogous sequences */c=[6,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,600, isA155566(n) & print1(n","))

A155568 Intersection of A001481 inter A020670: N = a^2 + b^2 = c^2 + 7d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 25, 29, 32, 36, 37, 49, 53, 64, 72, 81, 100, 109, 113, 116, 121, 128, 137, 144, 148, 149, 169, 193, 196, 197, 200, 212, 225, 232, 233, 256, 261, 277, 281, 288, 289, 296, 317, 324, 333, 337, 361, 373, 389, 392, 400, 401, 421, 424, 436, 441, 449

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155578 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155568(n,/* use optional 2nd arg to get other analogous sequences */c=[7,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,500, isA155568(n) & print1(n","))

A155570 Intersection of A003136 and A020669: N = a^2 + 3b^2 = c^2 + 5d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 9, 16, 21, 25, 36, 49, 61, 64, 81, 84, 100, 109, 121, 129, 144, 169, 181, 189, 196, 201, 225, 229, 241, 244, 256, 289, 301, 309, 324, 336, 349, 361, 381, 400, 409, 421, 436, 441, 469, 484, 489, 516, 525, 529, 541, 549, 576, 601, 625, 661, 669, 676, 709

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155710 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155570(n,/* use optional 2nd arg to get other analogous sequences */c=[5,3]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,800, isA155570(n) & print1(n","))

A301858 Positive integers which can be written as the sum of two squares but cannot be written as x^2 + y^2 + 2*z^2 with x and y integers and z a nonzero integer.

Original entry on oeis.org

1, 5, 29, 65

Offset: 1

Zhi-Wei Sun, Mar 27 2018

nonn

The sequence has no term in the interval [66, 10^6].
Conjecture 1: The sequence only has the four terms 1, 5, 29 and 65.
Conjecture 2: For any integer n > 1 which is neither 17 nor a power of 2, if n = u^2 + 2*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with z nonzero.
Conjecture 3: For any positive integer n not of the form 4^k*m (k = 0,1,2,... and m = 1, 7, 13), if n = u^2 + 3*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with y nonzero.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4

Cf. A000290, A000549, A001481, A002479, A301471.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[If[QQ[m]==False,Goto[aa]];Do[If[SQ[m-2x^2-y^2],Goto[aa]],{x,1,Sqrt[m/2]},{y,0,Sqrt[(m-2x^2)/2]}];tab=Append[tab,m];Label[aa],{m,1,1000}];Print[tab]

A034029 Numbers that are primitively represented by (x^2+2y^2 with x >= y >= 0).

Original entry on oeis.org

1, 3, 6, 11, 17, 18, 27, 33, 34, 38, 43, 51, 57, 66, 67, 81, 82, 83, 86, 89, 99, 102, 113, 114, 118, 121, 123, 129, 131, 139, 146, 153, 162, 171, 177, 179, 187, 193, 194, 198, 201, 209, 214, 219, 227, 233, 241, 242, 246, 249, 257, 258, 262, 267

Offset: 1

N. J. A. Sloane

nonn

Cf. A034027, A002479, A057127.

Definition corrected by N. J. A. Sloane, Apr 30 2015

A034032 Imprimitively but not primitively represented by x^2+2y^2.

Original entry on oeis.org

0, 4, 8, 12, 16, 24, 25, 32, 36, 44, 48, 49, 50, 64, 68, 72, 75, 76, 88, 96, 98, 100, 108, 128, 132, 136, 144, 147, 150, 152, 164, 169, 172, 176, 192, 196, 200, 204, 216, 225, 228, 236, 256, 264, 268, 272, 275, 288, 292, 294

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034030-A034034.

Corrected by N. J. A. Sloane, Apr 30 2015

A156381 Number of 3 X 3 arrays of squares of integers, symmetric about both diagonal and antidiagonal, with all rows summing to n.

Original entry on oeis.org

1, 2, 2, 1, 2, 0, 2, 0, 2, 4, 0, 2, 1, 0, 0, 0, 2, 2, 4, 2, 0, 0, 2, 0, 2, 4, 0, 3, 0, 0, 0, 0, 2, 4, 4, 0, 4, 0, 2, 0, 0, 4, 0, 2, 2, 0, 0, 0, 1, 2, 4, 5, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 0, 6, 2, 2, 0, 0, 0, 4, 2, 0, 3, 2, 0, 0, 0, 0, 8, 2, 2, 0, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 2, 7, 4, 0, 4, 0, 0, 0, 0, 2

Offset: 0

R. H. Hardin Feb 09 2009

nonn

a(n) is nonzero if and only if n = x^2 + 2*y^2 for some integers x and y if and only if n is in A002479. - Michael Somos, Dec 15 2011

			All solutions for n=9
...0.0.9...1.4.4...4.4.1...9.0.0
...0.9.0...4.1.4...4.1.4...0.9.0
...9.0.0...4.4.1...1.4.4...0.0.9

Cf. A002479.

cp's OEIS Frontend

A155565 Intersection of A001481 and A020669: N = a^2 + b^2 = c^2 + 5d^2 for some integers a,b,c,d.

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A155566 Intersection of A001481 and A002481: N = a^2 + b^2 = c^2 + 6d^2 for some integers a,b,c,d.

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A155568 Intersection of A001481 inter A020670: N = a^2 + b^2 = c^2 + 7d^2 for some integers a,b,c,d.

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PARI

A155570 Intersection of A003136 and A020669: N = a^2 + 3b^2 = c^2 + 5d^2 for some integers a,b,c,d.

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PARI

A301858 Positive integers which can be written as the sum of two squares but cannot be written as x^2 + y^2 + 2*z^2 with x and y integers and z a nonzero integer.

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Mathematica

A034029 Numbers that are primitively represented by (x^2+2y^2 with x >= y >= 0).

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A034032 Imprimitively but not primitively represented by x^2+2y^2.

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A156381 Number of 3 X 3 arrays of squares of integers, symmetric about both diagonal and antidiagonal, with all rows summing to n.

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