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 1-9 of 9 results.

A131851 Real part of the function z(n)=Sum(d(k)*i^k: d as in n=Sum(d(k)*2^k), i=sqrt(-1)).

Original entry on oeis.org

0, 1, 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, -1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, -1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, -1, 0, -1, 0, -2, -1, -2, -1, -1, 0, -1, 0, -2, -1, -2, -1, 0, 1, 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, -1, 0, -1, 0, -1, 0, -2, -1, -2, -1, -1, 0, -1, 0, -2, -1, -2
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

A131852(n) = Im(z(n));
z(A000079(n))=(A056594(n),A056594(n+3)); a(A000079(n))=A056594(n);
a(A131854(n))=0; a(A131861(n))>0; a(A131859(n))=1; a(A131863(n))<0;
z(A131853(n))=(0,0); z(A131856(n))=(0,1); z(A131858(n))=(1,0); z(A131860(n))=(1,1);
for n>0: a(A131865(n))=n and ABS(a(m))A131865(n).

Links

Crossrefs

Cf. A007088.

Programs

  • Mathematica
    z[0] = 0; z[n_] := z[n] = z[Floor[n/2]]*I + Mod[n, 2]; Table[z[n] // Re, {n, 0, 110}] (* Jean-François Alcover, Jul 03 2013 *)

Formula

z(n) = if n=0 then (0, 0) else z(floor(n/2))*(0, 1) + (n mod 2, 0), complex multiplication.

A131854 Numbers m such that A131851(m) = 0.

Original entry on oeis.org

0, 2, 5, 7, 8, 10, 13, 15, 20, 22, 28, 30, 32, 34, 37, 39, 40, 42, 45, 47, 52, 54, 60, 62, 65, 67, 73, 75, 80, 82, 85, 87, 88, 90, 93, 95, 97, 99, 105, 107, 112, 114, 117, 119, 120, 122, 125, 127, 128, 130, 133, 135, 136, 138, 141, 143, 148, 150, 156, 158, 160, 162, 165
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Links

Crossrefs

Cf. A131853, A131855, A131859, A131861, A131863.

Programs

  • Maple
    Z:= proc(n) option remember;
I*procname(floor(n/2))+(n mod 2)
end proc:
Z(0):= 0:
select(Re@Z=0, [$0..1000]); # Robert Israel, Dec 18 2017
  • Mathematica
    z[0] = 0; z[n_] := z[n] = z[Floor[n/2]]*I + Mod[n, 2]; Select[Range[0, 200], Re[z[#]] == 0&] (* Jean-François Alcover, Jan 31 2018 *)
  • PARI
    isok(n) = {d = Vecrev(binary(n)); real(sum(k=1, #d, d[k]*I^(k-1))) == 0;} \\ Michel Marcus, Jan 31 2018

A131855 Numbers m such that A131852(m) = 0.

Original entry on oeis.org

0, 1, 4, 5, 10, 11, 14, 15, 16, 17, 20, 21, 26, 27, 30, 31, 40, 41, 44, 45, 56, 57, 60, 61, 64, 65, 68, 69, 74, 75, 78, 79, 80, 81, 84, 85, 90, 91, 94, 95, 104, 105, 108, 109, 120, 121, 124, 125, 130, 131, 134, 135, 146, 147, 150, 151, 160, 161, 164, 165, 170, 171, 174
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Links

Crossrefs

Cf. A131852, A131853, A131854, A131857, A131862, A131864.

A131856 Numbers m such that z(m)=(0,1) with z as defined in A131851.

Original entry on oeis.org

2, 7, 22, 32, 37, 42, 47, 52, 62, 67, 82, 87, 97, 107, 112, 117, 122, 127, 162, 167, 182, 227, 242, 247, 262, 292, 302, 322, 327, 342, 352, 357, 362, 367, 372, 382, 422, 482, 487, 502, 512, 517, 522, 527, 532, 542, 552, 557, 572, 577, 587, 592, 597, 602, 607
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

Intersection of A131854 and A131857: A131851(a(n))=0, A131852(a(n))=1.

Links

Crossrefs

Cf. A131851, A131853, A131858, A131860.

A131858 Numbers m such that z(m)=(1,0) with z as defined in A131851.

Original entry on oeis.org

1, 11, 16, 21, 26, 31, 41, 56, 61, 81, 91, 121, 131, 146, 151, 161, 171, 176, 181, 186, 191, 211, 241, 251, 256, 261, 266, 271, 276, 286, 296, 301, 316, 321, 331, 336, 341, 346, 351, 361, 376, 381, 386, 391, 406, 416, 421, 426, 431, 436, 446, 451, 466, 471
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

Intersection of A131855 and A131859: A131851(a(n))=1, A131852(a(n))=0.

Links

Crossrefs

Cf. A131851, A131853, A131856, A131860.

A131860 Numbers m such that z(m)=(1,1) with z as defined in A131851.

Original entry on oeis.org

3, 18, 23, 33, 43, 48, 53, 58, 63, 83, 113, 123, 163, 178, 183, 243, 258, 263, 278, 288, 293, 298, 303, 308, 318, 323, 338, 343, 353, 363, 368, 373, 378, 383, 418, 423, 438, 483, 498, 503, 513, 523, 528, 533, 538, 543, 553, 568, 573, 593, 603, 633, 643, 658
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

Intersection of A131857 and A131859: A131851(a(n))=1, A131852(a(n))=1.

Links

Crossrefs

Cf. A131851, A131853, A131856, A131858.

A206715 Numbers matched to polynomials divisible by x^2+1.

Original entry on oeis.org

5, 10, 15, 20, 30, 40, 45, 60, 65, 75, 80, 85, 90, 95, 105, 120, 125, 130, 135, 150, 160, 165, 170, 175, 180, 190, 195, 210, 215, 225, 235, 240, 245, 250, 255, 260, 270, 300, 320, 325, 330, 335, 340, 350, 360, 365, 380, 390, 420, 430, 450, 455, 470
Offset: 1

Views

Author

Clark Kimberling, Feb 11 2012

Keywords

Comments

Is this a duplicate of A131853?
The polynomials having coefficients in {0,1} are enumerated as in A206074. The sequence A206715 shows the numbers of those satisfying p(n,i)=0.

Crossrefs

Cf. A131853, A206074, A206716.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], Factor[p[n, x]]},
  {n, 1, 16}]]
u = {}; Do[n++; If[(p[n, x] /. x -> I) == 0,
  AppendTo[u, n]], {n, 800}]
u    (* A206715 *)
u/5  (* A206716 *)

A206716 (1/5)A206715.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 15, 16, 17, 18, 19, 21, 24, 25, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 42, 43, 45, 47, 48, 49, 50, 51, 52, 54, 60, 64, 65, 66, 67, 68, 70, 72, 73, 76, 78, 84, 86, 90, 91, 94, 96, 97, 98, 99, 100, 102, 104, 105, 108, 117, 120, 121, 128
Offset: 1

Views

Author

Clark Kimberling, Feb 11 2012

Keywords

Comments

It is conjectured that all the terms of this sequence are integers; this may be equivalent to the conjecture at A131853.

Crossrefs

Cf. A131853, A206074, A206715.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], Factor[p[n, x]]},
  {n, 1, 16}]]
u = {}; Do[n++; If[(p[n, x] /. x -> I) == 0,
  AppendTo[u, n]], {n, 800}]
u    (* A206715 *)
u/5  (* A206716 *)

A330714 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let h(n) = Sum_{k=0..w} b_k * i^k (where i denotes the imaginary unit); a(n) is the square of the modulus of h(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 2, 1, 1, 2, 0, 1, 2, 1, 1, 0, 1, 4, 2, 5, 0, 1, 1, 2, 2, 5, 1, 4, 1, 2, 0, 1, 1, 2, 4, 5, 2, 1, 5, 4, 0, 1, 1, 2, 1, 0, 2, 1, 2, 5, 5, 8, 1, 2, 4, 5, 1, 4, 2, 5, 0, 1, 1, 2, 1, 0, 2, 1, 4, 1, 5, 2, 2, 1, 1, 0, 5, 2, 4, 1, 0, 1, 1, 2, 1
Offset: 0

Views

Author

Seiichi Manyama, Dec 27 2019

Keywords

Links

Crossrefs

Cf. A007088, A131851, A131852, A131853, A131856, A131858, A131860, A290886, A318479.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = a[Floor[n/2]]*I + Mod[n, 2]; Table[Abs[a[n]]^2, {n, 0, 100}] (* Amiram Eldar, May 06 2021, after Jean-François Alcover at A131851 *)
  • PARI
    {a(n) = my(d=Vecrev(digits(n, 2))); norm(sum(k=1, #d, d[k]*I^k))}

Formula

a(n) = A131851(n)^2 + A131852(n)^2.
Showing 1-9 of 9 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.