A184587 -id:A184587 - cp's OEIS frontend

A184586 a(n) = floor((n-1/2)*r), where r=sqrt(5); complement of A184587.

1, 3, 5, 7, 10, 12, 14, 16, 19, 21, 23, 25, 27, 30, 32, 34, 36, 39, 41, 43, 45, 48, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 74, 77, 79, 81, 83, 86, 88, 90, 92, 95, 97, 99, 101, 103, 106, 108, 110, 112, 115, 117, 119, 121, 124, 126, 128, 130, 133, 135, 137, 139, 141, 144, 146, 148, 150, 153, 155, 157, 159, 162, 164, 166, 168, 171, 173, 175, 177, 180, 182, 184, 186, 188, 191, 193, 195, 197, 200, 202, 204, 206, 209, 211, 213, 215, 218, 220, 222, 224, 226, 229, 231, 233, 235, 238, 240, 242, 244, 247, 249, 251, 253, 256, 258, 260, 262, 264, 267

Offset: 1

Clark Kimberling, Jan 17 2011

nonn

r = sqrt(5) and s = (5+sqrt(5))/4 form a Beatty pair. This yields the pair of complementary homogeneous Beatty sequences A022839 and A108598. From a theorem of Thoralf Skolem follows that (a(n)) and A184587 are complementary inhomogeneous Beatty sequences. - Michel Dekking, Sep 08 2017

Cf. A184587.

r=5^(1/2); c=1/2; s=r/(r-1);
Table[Floor[n*r-c*r],{n,1,120}]  (* A184586 *)
Table[Floor[n*s+c*s],{n,1,120}]  (* A184587 *)

a(n)=floor[(n-1/2)r], where r=sqrt(5).

Name and formula corrected by Michel Dekking, Sep 08 2017

A077545 Primes of the form floor(k*e).

Original entry on oeis.org

2, 5, 13, 19, 29, 43, 59, 67, 73, 89, 97, 103, 127, 149, 157, 163, 173, 179, 233, 239, 241, 263, 269, 271, 277, 293, 307, 331, 337, 347, 353, 383, 421, 443, 467, 521, 557, 587, 617, 619, 641, 701, 709, 733, 739, 761, 769, 823, 829, 839, 853, 883, 907, 929, 937

Offset: 1

Amarnath Murthy, Nov 09 2002

Primes not in A077545 are in A184856, since {floor(k*e)} and {floor(j*e/(e-1))} are complementary Beatty sequences (A022843 and A054385).

Cf. A062409, A184855, A184856, A184587, A184858.

r=E; s=r/(r-1);
a[n_]:=Floor[n*r];
b[n_]:=Floor[n*s];
Table[a[n], {n, 1, 120}]  (* A022843 *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1, a[n]]], {n, 1, 600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2, n]], {n, 1, 600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3, n]], {n, 1, 300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4, b[n]]], {n, 1, 600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5, n]], {n, 1, 600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6, n]], {n, 1, 300}]; t6
(* List t1 matches A077545; list t2 matches A062409;
lists t3-t6 match A184855-A184858. *)

More terms from Sascha Kurz, Jan 12 2003

Mathematica code and crossreferences by Clark Kimberling, Jan 24 2011

cp's OEIS Frontend

A184586 a(n) = floor((n-1/2)*r), where r=sqrt(5); complement of A184587.

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