cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A270050 Numbers of the form 2 * (x^2 + xy + y^2).

Original entry on oeis.org

0, 2, 6, 8, 14, 18, 24, 26, 32, 38, 42, 50, 54, 56, 62, 72, 74, 78, 86, 96, 98, 104, 114, 122, 126, 128, 134, 146, 150, 152, 158, 162, 168, 182, 186, 194, 200, 206, 216, 218, 222, 224, 234, 242, 248, 254, 258, 266, 278, 288, 294, 296, 302, 312, 314, 326, 338, 342, 344
Offset: 1

Views

Author

Altug Alkan, Mar 09 2016

Keywords

Comments

Integers of the form (x^2 + xy + y^2) / 2. See comments in A266836 about the numbers of the form x^2 + xy + y^2.

Examples

			6 is a term because 6 = (4^2 + 4*(-2) + (-2)^2) / 2.

Links

Crossrefs

Cf. A003136.

Programs

  • Mathematica
    Select[Range[0, 400], Resolve@ Exists[{x, y}, Reduce[# == (x^2 + x y + y^2)/2, {x, y}, Integers]] &] (* Michael De Vlieger, Mar 09 2016 *)
  • PARI
    x='x+O('x^700); p=eta(x)^3/eta(x^3); for(n=0, 699, if(polcoeff(p, n) != 0 && n % 2 == 0, print1(n/2, ", ")));
  • PARI
    list(lim)=my(v=List(), y, t); lim\=2; for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, listput(v, 2*t); y++)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017

Formula

a(n) = 2 * A003136(n).

A270248 Even Löschian numbers.

Original entry on oeis.org

0, 4, 12, 16, 28, 36, 48, 52, 64, 76, 84, 100, 108, 112, 124, 144, 148, 156, 172, 192, 196, 208, 228, 244, 252, 256, 268, 292, 300, 304, 316, 324, 336, 364, 372, 388, 400, 412, 432, 436, 444, 448, 468, 484, 496, 508, 516, 532, 556, 576, 588, 592, 604, 624, 628, 652, 676
Offset: 1

Views

Author

Altug Alkan, Mar 14 2016

Keywords

Comments

Even numbers of the form x^2 - xy + y^2.

Examples

			Even number 12 is a term because 12 = 2^2 + 2*2 + 2^2.

Links

Crossrefs

Cf. Loeschian numbers: A003136 (all), A266836 (2*k+1), A202822 (3*k+1), A260682 (6*k+1).

Programs

  • Mathematica
    Select[Range[0, 680, 2], Resolve@ Exists[{x, y}, Reduce[# == (x^2 - x y + y^2), {x, y}, Integers]] &] (* Michael De Vlieger, Mar 15 2016 *)
  • PARI
    x='x+O('x^800); p=eta(x)^3/eta(x^3); for(n=0, 799, if(polcoeff(p, n) != 0 && n % 2 == 0, print1(n, ", ")));
  • PARI
    list(lim)=my(v=List(), y, t); forstep(x=0, sqrtint(lim\3), 2, my(y=x, t); while((t=x^2+x*y+y^2)<=lim, listput(v, t); y+=2)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017

Formula

a(n) = 2 * A270050(n) = 4 * A003136(n).

A266918 Perfect power Löschian numbers.

Original entry on oeis.org

1, 4, 9, 16, 25, 27, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2116, 2187, 2197, 2209, 2304, 2401, 2500
Offset: 1

Views

Author

Altug Alkan, Jan 06 2016

Keywords

Comments

Inspired by A266836. See first comment in A266836.
Intersection of A001597 and A003136.
Obviously, this sequence contains all positive squares.
Perfect powers that are not the Löschian numbers are 8, 32, 125, 128, 216, 512, 1000, 1331, 2048, 2744, 3125, 3375, 4913, 5832, 7776, ...

Examples

			25 is a term because 25 = 5^2 = 5^2 + 5*0 + 0^2.
27 is a term because 27 = 3^3 = 3^2 + 3*3 + 3^2.
243 is a term because 243 = 3^5 = 9^2 + 9*9 + 9^2.
343 is a term because 343 = 7^3 = 18^2 + 18*1 + 1^2.

Links

Crossrefs

Cf. Loeschian numbers: A003136 (all), A266836 (2*k+1), A202822 (3*k+1), A260682 (6*k+1).
Cf. A001597.

Programs

  • Mathematica
    fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; gQ[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x y + y^2, {x, y}, Integers]]]; Select[Range@ 2500, fQ@# && gQ@# &] (* Michael De Vlieger, Jan 06 2016, after Ant King at A001597 and Jean-François Alcover at A003136 *)
  • PARI
    x='x+O('x^10^4); p=eta(x)^3/eta(x^3); for(n=0, 9999, if(polcoeff(p, n) != 0 && (ispower(n) || n==1), print1(n, ", ")));
  • PARI
    is(n) = (ispower(n) || n==1) && #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e4, if(is(n), print1(n, ", ")));

A270672 Löschian numbers (A003136) that are multiples of 3.

Original entry on oeis.org

0, 3, 9, 12, 21, 27, 36, 39, 48, 57, 63, 75, 81, 84, 93, 108, 111, 117, 129, 144, 147, 156, 171, 183, 189, 192, 201, 219, 225, 228, 237, 243, 252, 273, 279, 291, 300, 309, 324, 327, 333, 336, 351, 363, 372, 381, 387, 399, 417, 432, 441, 444, 453, 468, 471, 489, 507, 513, 516, 525, 543, 549
Offset: 1

Views

Author

Altug Alkan, Mar 21 2016

Keywords

Comments

Numbers of the form 3*(x^2 + xy + y^2).
Intersection of A008585 and A003136.

Examples

			21 is a term because 21 = 3*7 = 4^2 + 4*1 + 1^2.

Links

Crossrefs

Cf. Loeschian numbers: A003136 (all), A270248 (2*k), A266836 (2*k+1), A202822 (3*k+1), A260682 (6*k+1).

Programs

  • Mathematica
    Select[Range[0, 600], Resolve[Exists[{x, y}, Reduce[# == 3 (x^2 + x y + y^2), {x, y}, Integers]]] &] (* Michael De Vlieger, Mar 21 2016 *)
  • PARI
    x='x+O('x^1000); p=eta(x)^3/eta(x^3); for(n=0, 999, if(polcoeff(p, n) != 0 && n % 3 == 0, print1(n, ", ")));
  • PARI
    list(lim)=my(v=List(), y, t); for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, listput(v, t); y+=3)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.