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A190304 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=csch(1), t=sech(1). 3
1, 4, 6, 9, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 39, 42, 44, 47, 49, 51, 54, 56, 59, 62, 64, 66, 69, 71, 74, 77, 79, 82, 84, 86, 89, 91, 94, 97, 99, 101, 104, 106, 109, 112, 114, 116, 119, 121, 124, 127, 129, 132, 133, 136, 139, 141, 144, 147, 149, 151, 154, 156, 159, 162, 164, 167, 169, 171, 174, 177, 179, 182, 183, 186, 189 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that

a(n) = n+[ns/r]+[nt/r],

b(n) = n+[nr/s]+[nt/s],

c(n) = n+[nr/t]+[ns/t], where []=floor.

Taking r=1, s=csch(1), t=sech(1) gives

a=A190304, b=A190305, c=A190306.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

FORMULA

A190304:  a(n)=n+[n*csch(1)]+[n*sech(1)].

A190305:  b(n)=n+[n*sinh(1)]+[n*tanh(1)].

A190306:  c(n)=n+[n*cosh(1)]+[n*coth(1)].

MATHEMATICA

r=1; s=Csch[1]; t=Sech[1];

a[n_] := n + Floor[n*s/r] + Floor[n*t/r];

b[n_] := n + Floor[n*r/s] + Floor[n*t/s];

c[n_] := n + Floor[n*r/t] + Floor[n*s/t];

Table[a[n], {n, 1, 120}]  (*A190304*)

Table[b[n], {n, 1, 120}]  (*A190305*)

Table[c[n], {n, 1, 120}]  (*A190306*)

PROG

(PARI) for(n=1, 30, print1(n + floor(n/sinh(1)) + floor(n/cosh(1)), ", ")) \\ G. C. Greubel, Dec 29 2017

(MAGMA) [n + Floor(n/Sinh(1)) + Floor(n/Cosh(1)): n in [1..30]]; // G. C. Greubel, Dec 29 2017

CROSSREFS

Cf. A190305, A190306, A190265.

Sequence in context: A292660 A190081 A298468 * A189366 A066095 A003622

Adjacent sequences:  A190301 A190302 A190303 * A190305 A190306 A190307

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 07 2011

STATUS

approved

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Last modified November 18 20:13 EST 2018. Contains 317324 sequences. (Running on oeis4.)