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.

A063672 Sequence A019320 in binary.

Original entry on oeis.org

10, 1, 11, 111, 101, 11111, 11, 1111111, 10001, 1001001, 1011, 11111111111, 1101, 1111111111111, 101011, 10010111, 100000001, 11111111111111111, 111001, 1111111111111111111, 11001101, 100100110111, 1010101011
Offset: 0

Views

Author

Antti Karttunen, Aug 03 2001

Keywords

Links

Crossrefs

Cf. A063697, A063699, A063671.

Programs

  • Maple
    map(convert, A019320,binary); or up to n=104 with: map(convert,[seq(Phi_pos_terms(j,2)-Phi_neg_terms(j,2),j=0..104)],binary);
  • Mathematica
    A063672[n_] := If[n == 0, 10, FromDigits[IntegerDigits[Cyclotomic[n, 2], 2]]];
Array[A063672, 30, 0] (* Paolo Xausa, Feb 26 2024 *)

A063697 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), A063696 in binary.

Original entry on oeis.org

0, 10, 11, 111, 101, 11111, 101, 1111111, 10001, 1001001, 10101, 11111111111, 10001, 1111111111111, 1010101, 100101001, 100000001, 11111111111111111, 1000001, 1111111111111111111, 100010001, 1001001001001, 10101010101
Offset: 0

Views

Author

Antti Karttunen, Aug 03 2001

Keywords

Examples

			Phi_15(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1, thus the 1-bits of a(15) are at positions 0,3,5 and 8, thus we get a(15) = 100101001

Links

Crossrefs

Cf. A063699, A063671, A063672.

Programs

  • Maple
    map(convert, A063696,binary);

Extensions

a(0) corrected by Sean A. Irvine, May 08 2023

A063699 Positions of negative coefficients in cyclotomic polynomial Phi_n(x), A063698 in binary.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 10, 0, 0, 0, 1010, 0, 100, 0, 101010, 10010010, 0, 0, 1000, 0, 1000100, 100100010010, 1010101010, 0, 10000, 0, 101010101010, 0, 10001000100, 0, 111000, 0, 0, 10010010010010010010, 1010101010101010, 100001010010100101000010
Offset: 0

Views

Author

Antti Karttunen, Aug 03 2001

Keywords

Examples

			E.g. Phi_15(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1, thus the 1-bits of a(15) are at positions 1,4 and 7, thus we get a(15) = 10010010

Links

Crossrefs

Cf. A063697, A063671, A063672.

Programs

  • Maple
    map(convert, A063698,binary);

A063670 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

Original entry on oeis.org

2, 3, 3, 7, 5, 31, 7, 127, 17, 73, 31, 2047, 21, 8191, 127, 443, 257, 131071, 73, 524287, 341, 7003, 2047, 8388607, 273, 1082401, 8191, 262657, 5461, 536870911, 443, 2147483647, 65537, 1797851, 131071, 26181091, 4161, 137438953471, 524287
Offset: 0

Views

Author

Antti Karttunen, Aug 03 2001

Keywords

Comments

a(n) = 2^n-1 whenever n is prime. It seems as if a(n) >= A005420(n) for all n (checked up to 200), with equality for all 1A005420(n)=2^n-1 (i.e., 2^n-1 is prime). - M. F. Hasler, Apr 30 2007
a(0) could also be 1. - T. D. Noe, Oct 29 2007

Links

Crossrefs

Cf. A013594.
a(n) = A063696(n) (the positive terms) + A063698(n) (the negative terms).
This sequence in binary: A063671.
Cf. A005420.

Programs

  • Maple
    [seq(Phi_pos_terms(j,2)+Phi_neg_terms(j,2),j=0..104)];
  • Mathematica
    a[n_] := FromDigits[ If[# != 0, 1, 0]& /@ CoefficientList[ Cyclotomic[n, x], x], 2]; a[0] = 2; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Dec 11 2012 *)
  • PARI
    A063670(n)=local(p=polcyclo(n+!n)); if(n,sum(i=0, n, (polcoeff(p, i)<>0)<M. F. Hasler, Apr 30 2007
  • PARI
    a(n) = subst(apply(x->x!=0, polcyclo(n,'x)), 'x, 2);  \\ Gheorghe Coserea, Nov 04 2016
Showing 1-4 of 4 results.

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