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.

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A152149 Decimal expansion of the angle B in the doubly golden triangle ABC.

Original entry on oeis.org

6, 5, 7, 4, 0, 5, 4, 8, 2, 9, 7, 6, 5, 3, 2, 5, 9, 2, 3, 8, 0, 9, 6, 8, 5, 4, 1, 5, 2, 9, 3, 9, 7, 1, 2, 6, 5, 4, 1, 4, 9, 5, 9, 4, 6, 4, 8, 7, 8, 3, 9, 3, 7, 0, 7, 8, 2, 0, 9, 2, 8, 0, 8, 5, 8, 8, 5, 3, 9, 5, 0, 6, 1, 3, 8, 1, 7, 7, 3, 5, 0, 7, 0, 1, 7, 1, 5, 1, 6, 5, 4, 4, 0, 5, 2, 2, 7, 8, 0, 5, 2, 8, 1, 2, 6
Offset: 0

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Author

Clark Kimberling, Nov 26 2008

Keywords

Comments

There is a unique (shape of) triangle ABC that is both side-golden and angle-golden. Its angles are B, C=t*B and A=pi-B-t*B, where t is the golden ratio. "Angle-golden" and "side-golden" refer to partitionings of ABC, each in a manner that matches the continued fraction [1,1,1,...] of t. (The partitionings are analogous to the partitioning of the golden rectangle into squares by the removal of exactly 1 square at each stage.)
For doubly silver and doubly e-ratio triangles, see A188543 and A188544.
For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.

Examples

			The number B begins with 0.65740548 (equivalent to 37.666559... degrees).

References

  • Clark Kimberling, "A new kind of golden triangle," in Applications of Fibonacci Numbers, Proc. Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1991.

Links

Crossrefs

Cf. A000045, A188543, A188544, A376961 (length of shortest side of the doubly golden triangle that has area 1).

Programs

  • Mathematica
    r = (1 + 5^(1/2))/2; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]
  • PARI
    t=(1+5^(1/2))/2; solve(b=.6, .7, sin(b*t^2)-t*sin(b)) \\ Iain Fox, Feb 11 2020

Formula

B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=(1+5^(1/2))/2, the golden ratio.

Extensions

Keyword:cons added and offset corrected by R. J. Mathar, Jun 18 2009

A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 7, 18, 27, 8, 11, 21, 54, 81, 10, 12, 33, 63, 162, 243, 13, 14, 36, 99, 189, 486, 729, 16, 15, 42, 108, 297, 567, 1458, 2187, 17, 19, 45, 126, 324, 891, 1701, 4374, 6561, 20, 22, 49, 135, 378, 972, 2673, 5103, 13122, 19683, 25, 23, 57, 147, 405, 1134, 2916, 8019, 15309, 39366, 59049, 26, 24, 66, 171, 441, 1215, 3402, 8748, 24057, 45927, 118098, 177147
Offset: 1

Views

Author

Clark Kimberling, Jan 14 2025

Keywords

Comments

Every positive integer occurs exactly once.

Examples

			Corner:
      1     2     4     5     8     10     13     16      17
      3     6     7    11    12     14     15     19      22
      9    18    21    33    36     42     45     49      57
     27    54    63    99   108    126    135    147     171
     81   162   189   297   324    378    405    441     513
    243   486   567   891   972   1134   1215   1323    1539
    729  1458  1701  2673  2916   3402   3645   3969    4617
   2187  4374  5103  8019  8748  10206  10935  11907   13851

Crossrefs

Cf. A065339, A002144, A002145, A376961, A378193, A072437 (row 1), A000244 (column 0), A025192 (column 1).

Programs

  • Maple
    A378194 := proc(n, k)
    option remember;
    local a;
    if k = 0 then
        0;
    else
        for a from procname(n, k-1)+1 do
            if A065339(a) = n-1 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(seq( A378194(n, d-n), n=1..d-1), d=2..10) ; # R. J. Mathar, Jan 28 2025
  • Mathematica
    u = Map[Map[#[[1]] &, #] &, GatherBy[
    SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 3)/4] && PrimeQ[#] &,
             Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
             FactorInteger[#]]]], True]} &,
      Range[24000]], #[[2]] &], #[[2]] &]];
r[m_] := Take[u[[m]], 10];
w[m_, n_] := r[m][[n]];
Grid[Table[w[m, n], {m, 1, 8}, {n, 1, 9}]]   (* array *)
Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Nov 19 2024 *)

Extensions

Definition corrected. - R. J. Mathar, Jan 28 2025

A380657 Numbers whose prime factorization has more Pythagorean prime factors than non-Pythagorean prime factors (including multiplicities).

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 50, 53, 61, 65, 73, 75, 85, 89, 97, 101, 109, 113, 125, 130, 137, 145, 149, 157, 169, 170, 173, 175, 181, 185, 193, 195, 197, 205, 221, 229, 233, 241, 250, 255, 257, 265, 269, 275, 277, 281, 289, 290, 293, 305, 313, 317, 325, 337
Offset: 1

Views

Author

Clark Kimberling, Jan 30 2025

Keywords

Examples

			50 appears because 2*5*5 has 2 Pythagorean prime factors but only 1 non-Pythagorean prime factor.

Crossrefs

Cf. A000040, A002144, A002145, A083025, A376961, A377625, A378137, A378139, A378140.

Programs

  • Mathematica
    f[{x_, y_}] := If[Mod[x, 4] == 1, y, -y];
s[n_] := Map[f, FactorInteger[n]];
p[n_] := {Total[Select[s[n], # > 0 &]], -Total[Select[s[n], # < 0 &]]};
p[1] = {0, 0};
t = Table[p[n], {n, 1, 500}];
u = Map[First, t];  (* A083025 *)
v = Map[Last, t] ;  (* A376961 *)
v - u (* A377625 *);
Flatten[Position[v - u, -1]]  (* this sequence *)
Showing 1-3 of 3 results.

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