A227631 -id:A227631 - cp's OEIS frontend

A227776 a(n) = 6*n^2 + 1.

1, 7, 25, 55, 97, 151, 217, 295, 385, 487, 601, 727, 865, 1015, 1177, 1351, 1537, 1735, 1945, 2167, 2401, 2647, 2905, 3175, 3457, 3751, 4057, 4375, 4705, 5047, 5401, 5767, 6145, 6535, 6937, 7351, 7777, 8215, 8665, 9127, 9601, 10087, 10585, 11095, 11617, 12151

Offset: 0

Clark Kimberling, Jul 30 2013

Least splitter is defined for x < y at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Conjecture: a(n) is the least splitter of s(n) and s(n+1), where s(n) = n*sin(1/n).

			The first eight least splitting rationals for {n*sin(1/n), n >=1 } are these fractions: 6/7, 24/25, 54/55, 96/97, 150/151, 216/217, 294/295, 384/385.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Hexagonal Star Rays.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A227631, A287326.

Cf. A003154, A008458, A003215, A000217, A028896, A054554, A046092, A080855, A045943, A172043, A002378, A033581.

z = 40; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = n*Sin[1/n]; t = Table[r[s[n], s[n + 1]], {n, 1, z}] (* least splitting rationals *); fd = Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)
Array[6 #^2 + 1 &, 45] (* Michael De Vlieger, Nov 08 2017 *)
LinearRecurrence[{3,-3,1},{7,25,55},50] (* Harvey P. Dale, Dec 16 2017 *)

a(n)=6*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (1 + 4*x + 7*x^2)/(1 - x)^3.
a(n) = A287326(2n, n). - Kolosov Petro, Nov 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(6))*coth(Pi/sqrt(6)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(6))*csch(Pi/sqrt(6)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(6))*sinh(Pi/sqrt(3)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(6))*csch(Pi/sqrt(6)).(End)
From Leo Tavares, Nov 20 2021: (Start)
a(n) = A003154(n+1) - A008458(n). See Hexagonal Star Rays illustration.
a(n) = A003215(n) + A028896(n-1).
a(n) = A054554(n+1) + A046092(n).
a(n) = A080855(n) + A045943(n).
a(n) = A172043(n) + A002378(n).
a(n) = A033581(n) + 1. (End)
E.g.f.: exp(x)*(1 + 6*x + 6*x^2). - Stefano Spezia, Sep 14 2024

a(0) = 1 prepended by Robert P. P. McKone, Oct 09 2023

A227629 Least splitter of the harmonic numbers H(n) and H(n+1).

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 3, 4, 6, 1, 10, 6, 4, 7, 3, 5, 9, 2, 7, 5, 3, 7, 4, 5, 6, 7, 10, 14, 27, 1, 18, 12, 9, 7, 6, 5, 9, 4, 11, 7, 13, 3, 11, 8, 5, 7, 9, 13, 25, 2, 15, 9, 7, 12, 5, 8, 11, 17, 3, 13, 10, 7, 15, 4, 13, 9, 5, 11, 17, 6, 7, 15, 8, 9, 11, 12, 14, 17

Offset: 1

Clark Kimberling, Jul 18 2013

See A227631 for the definition of least splitter.

			The first few splitting rationals are 1/1, 3/2, 2/1, 9/4, 7/3, 5/2, 8/3, 11/4, 17/6, 3/1, 31/10, 19/6; e.g. 9/4 splits H(4) and H(5), as indicated by H(4) = 1 + 1/2 + 1/3 + 1/4 =  2.083...  < 2.25 < 2.283... = H(5) and the chain H(1) <= 1/1 < H(2) < 3/2 < H(3) < 2/1 < H(4) < 9/4 < ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227630, A227631.

h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[h[n], h[n + 1]], {n, 1, 120}];
Denominator[t] (* A227629 *)
Numerator[t]   (* A227630 *)  (* Peter J. C. Moses, Jul 15 2013 *)

A227630 Numerator of the least splitting rational of the harmonic numbers H(n) and H(n+1).

Original entry on oeis.org

1, 3, 2, 9, 7, 5, 8, 11, 17, 3, 31, 19, 13, 23, 10, 17, 31, 7, 25, 18, 11, 26, 15, 19, 23, 27, 39, 55, 107, 4, 73, 49, 37, 29, 25, 21, 38, 17, 47, 30, 56, 13, 48, 35, 22, 31, 40, 58, 112, 9, 68, 41, 32, 55, 23, 37, 51, 79, 14, 61, 47, 33, 71, 19, 62, 43, 24

Offset: 1

Clark Kimberling, Jul 18 2013

See A227631 for the definition of least splitting rational.

			The first few splitting rationals are 1/1, 3/2, 2/1, 9/4, 7/3, 5/2, 8/3, 11/4, 17/6, 3/1, 31/10, 19/6; e.g. 9/4 splits H(4) and H(5), as indicated by H(4) = 1 + 1/2 + 1/3 + 1/4 =  2.083...  < 2.25 < 2.283... = H(5) and the chain H(1) <= 1/1 < H(2) < 3/2 < H(3) < 2/1 < H(4) < 9/4 < ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227629, A227631.

h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[h[n], h[n + 1]], {n, 1, 120}];
Denominator[t] (* A227629 *)
Numerator[t]   (* A227630 *)  (* Peter J. C. Moses, Jul 15 2013 *)

A227687 Least splitter of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 2, 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 5, 3, 2, 3, 1, 5, 3, 2, 3, 5, 1, 4, 3, 2, 3, 5, 1, 4, 3, 2, 3, 4, 1, 6, 4, 3, 2, 3, 5, 1, 6, 4, 3, 2, 3, 4, 7, 1, 5, 3, 5, 2, 3, 4, 7, 1, 5, 4, 3, 2, 3, 4, 6, 1, 7, 4, 3, 5, 2, 3, 4, 6, 1, 8, 5

Offset: 1

Clark Kimberling, Jul 21 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y.

The positions of 1 in this sequences (indicating those least splitting rationals of s(n) and s(n+1) which are integers) are given by A186351.

			The denominators (A227687) and numerators (A227688) can be read from these chains:
1 < 2 < 5/2 < 3 < 7/2 < 4 < 13/3 < 9/2 < 5 < 21/4 < 11/2 < 17/3 < 6 < . . .
s(1) <= 1 < s(2) < 2 < s(3) < 5/2 < s(4) < 3 < s(5) < 4 < s(6) < 13/3 <  . . .

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631, A227688.

r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d];
s[n_] := s[n] = Sum[k^(-1/2), {k, 1, n}]; t = Table[r[s[n], s[n + 1]], {n, 1, 15}] (*fractions*)
fd = Denominator[t] (*A227687*)
fn = Numerator[t]   (*A227688*)

A227688 Numerator of least splitting rational of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

Original entry on oeis.org

1, 2, 5, 3, 7, 4, 13, 9, 5, 21, 11, 17, 6, 19, 13, 20, 7, 22, 15, 23, 8, 41, 25, 17, 26, 9, 46, 28, 19, 29, 49, 10, 41, 31, 21, 32, 54, 11, 45, 34, 23, 35, 47, 12, 73, 49, 37, 25, 38, 64, 13, 79, 53, 40, 27, 41, 55, 97, 14, 71, 43, 72, 29, 44, 59, 104, 15

Offset: 1

Clark Kimberling, Jul 21 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y.

			The denominators (A227687) and numerators (A227688) can be read from these chains:
1 < 2 < 5/2 < 3 < 7/2 < 4 < 13/3 < 9/2 < 5 < 21/4 < 11/2 < 17/3 < 6 < . . . ;
s(1) <= 1 < s(2) < 2 < s(3) < 5/2 < s(4) < 3 < s(5) < 4 < s(6) < 13/3 <  . . .

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631, A227687.

r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d];
s[n_] := s[n] = Sum[k^(-1/2), {k, 1, n}]; t = Table[r[s[n], s[n + 1]], {n, 1, 15}] (*fractions*)
fd = Denominator[t] (*A227687*)
fn = Numerator[t]   (*A227688*)

A225594 Least splitter of s(n) and s(n+1), where s(n) = (1 + n)^(1/n).

Original entry on oeis.org

1, 3, 5, 9, 2, 11, 9, 7, 12, 5, 18, 13, 8, 19, 11, 25, 14, 17, 23, 26, 35, 44, 65, 116, 3, 115, 73, 55, 46, 37, 34, 31, 28, 25, 47, 22, 41, 19, 73, 35, 51, 83, 16, 61, 45, 29, 42, 55, 68, 107, 13, 101, 75, 49, 85, 36, 59, 105, 23, 79, 56, 33, 109, 76, 43, 53

Offset: 1

Clark Kimberling, Jul 30 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Since s(n) -> e, the least splitting rationals -> e.

			The first 15 splitting rationals are 2/1, 7/3, 12/5, 22/9, 5/2, 28/11, 23/9, 18/7, 31/12, 13/5, 47/18, 34/13, 21/8, 50/19, 29/11.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631.

z = 100; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = (1 + 1/n)^n; t = Table[r[s[n], s[n + 1]], {n, 1, z}]; fd = Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)

A227634 Least splitter of log(n) and log(n+1).

Original entry on oeis.org

1, 1, 3, 2, 3, 5, 1, 6, 4, 3, 5, 2, 5, 3, 4, 5, 6, 10, 18, 1, 11, 8, 6, 5, 4, 7, 10, 3, 5, 7, 9, 15, 2, 11, 7, 5, 8, 14, 3, 10, 7, 4, 9, 5, 11, 6, 7, 8, 10, 12, 15, 21, 34, 1, 40, 24, 17, 13, 11, 10, 8, 7, 13, 6, 11, 5, 14, 9, 17, 4, 11, 7, 10, 13, 22, 3, 17

Offset: 1

Clark Kimberling, Jul 18 2013

Essentially the same as A183163. - R. J. Mathar, Jul 27 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y.

			The splitting rationals of consecutive numbers log(1), log(2), ... are 0, 1, 4/3, 3/2, 5/3, 9/5, 2, 13/6, 9/4, 7/3, 12/5, 5/2, 13/5; the denominators form A227634, and the numerators, A227684.  Chain:
log(1) <= 0 < log(2) < 1 < log(3) < 4/3 < log(4) < 3/2 < log(5) < 5/3 < ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631.

h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[Log[n], Log[n + 1]], {n, 1, 120}] (*fractions*)
Denominator[t] (* A227634 *)
Numerator[t]   (* A227684 *)

A227684 Numerator of least splitting rational of log(n) and log(n+1).

Original entry on oeis.org

0, 1, 4, 3, 5, 9, 2, 13, 9, 7, 12, 5, 13, 8, 11, 14, 17, 29, 53, 3, 34, 25, 19, 16, 13, 23, 33, 10, 17, 24, 31, 52, 7, 39, 25, 18, 29, 51, 11, 37, 26, 15, 34, 19, 42, 23, 27, 31, 39, 47, 59, 83, 135, 4, 161, 97, 69, 53, 45, 41, 33, 29, 54, 25, 46, 21, 59, 38

Offset: 1

Clark Kimberling, Jul 19 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y.

			The splitting rationals of consecutive numbers log(1), log(2), ... are 0, 1, 4/3, 3/2, 5/3, 9/5, 2, 13/6, 9/4, 7/3, 12/5, 5/2, 13/5; the denominators form A227634, and the numerators, A227684.  Chain:
log(1) <= 0 < log(2) < 1 < log(3) < 4/3 < log(4) < 3/2 < log(5) < 5/3 < ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631, A227634.

h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[Log[n], Log[n + 1]], {n, 1, 120}] (*fractions*)
Denominator[t] (* A227634 *)
Numerator[t]  (* A227684 *)

A227685 Least splitter of s(n) and s(n+1), where s(n) = 1 + 1/2^2 + ... + 1/n^2.

Original entry on oeis.org

1, 3, 5, 7, 15, 2, 19, 13, 11, 9, 16, 23, 7, 19, 12, 29, 17, 22, 32, 57, 5, 88, 53, 38, 28, 51, 23, 59, 18, 31, 44, 70, 13, 60, 47, 34, 76, 21, 50, 29, 66, 37, 45, 53, 69, 85, 117, 189, 8, 243, 147, 107, 83, 67, 59, 51, 94, 43, 78, 35, 97, 62, 89, 27, 154

Offset: 1

Clark Kimberling, Jul 19 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y.

			The denominators (A227685) and numerators (A227686) can be read from this chain: s(1) <= 1 < s(2) < 4/3 < s(3) < 7/5 < s(4) < 10/7 < s(5) < 22/15 < ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631, A227686.

Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = Sum[k^(-2), {k, 1, n}]
t = Table[r[s[n], s[n + 1]], {n, 1, 150}] (*fractions*)
Denominator[t] (*A227685*)
Numerator[t]   (*A227686*)

A227686 Numerator of least splitting rational of s(n) and s(n+1), where s(n) = 1 + 1/2^2 + ... + 1/n^2.

Original entry on oeis.org

1, 4, 7, 10, 22, 3, 29, 20, 17, 14, 25, 36, 11, 30, 19, 46, 27, 35, 51, 91, 8, 141, 85, 61, 45, 82, 37, 95, 29, 50, 71, 113, 21, 97, 76, 55, 123, 34, 81, 47, 107, 60, 73, 86, 112, 138, 190, 307, 13, 395, 239, 174, 135, 109, 96, 83, 153, 70, 127, 57, 158, 101

Offset: 1

Clark Kimberling, Jul 19 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y.

			The denominators (A227685) and numerators (A227686) can be read from this chain:  s(1) <= 1 < s(2) < 4/3 < s(3) < 7/5 < s(4) < 10/7 < s(5) < 22/15 < ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631, A227685.

Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = Sum[k^(-2), {k, 1, n}]
t = Table[r[s[n], s[n + 1]], {n, 1, 150}] (*fractions)
fd = Denominator[t] (*A227685*)
fn = Numerator[t]   (*A227686*)

cp's OEIS Frontend

A227776 a(n) = 6*n^2 + 1.

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A227629 Least splitter of the harmonic numbers H(n) and H(n+1).

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A227630 Numerator of the least splitting rational of the harmonic numbers H(n) and H(n+1).

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A227687 Least splitter of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

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A227688 Numerator of least splitting rational of s(n) and s(n+1), where s(n) = 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n).

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A225594 Least splitter of s(n) and s(n+1), where s(n) = (1 + n)^(1/n).

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A227634 Least splitter of log(n) and log(n+1).

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A227684 Numerator of least splitting rational of log(n) and log(n+1).

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