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.

A146309 a(n) = indices where primes occurred in A146306.

Original entry on oeis.org

1, 3, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 58, 62, 66, 74, 78, 82, 86, 94, 102, 106, 114, 118, 122, 134, 138, 142, 146, 158, 166, 174, 178, 186, 194, 202, 206, 214, 218, 222, 226, 246, 254, 258, 262, 274, 278, 282, 298, 302, 314, 318, 326, 334, 346, 354, 358
Offset: 0

Views

Author

Artur Jasinski, Oct 29 2008

Keywords

Comments

General formula (*Artur Jasinski*):
2 Cos[2*Pi/n] = Hypergeometric2F1[(n-6)/(2n),(n+6)/(2n),1/2,3/4] =
Hypergeometric2F1[A146306(n)/A146307(n),A146306(n+12)/A146307(n),1/2,3/4].
2 Cos[2*Pi/n] is root of polynomial of degree = EulerPhi[n]/2 = A000010(n)/2 = A023022(n).

Crossrefs

A007310, A051724, A146306, A146307, A146308.

Programs

  • Mathematica
    aa = {}; Do[k = Denominator[(n - 6)/(2 n)]; If[PrimeQ[k], AppendTo[aa, n]], {n, 1, 1000}]; aa (*Artur Jasinski*)

A146535 Numerator of (2*n-1)/3.

Original entry on oeis.org

1, 1, 5, 7, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9, 29, 31, 11, 35, 37, 13, 41, 43, 15, 47, 49, 17, 53, 55, 19, 59, 61, 21, 65, 67, 23, 71, 73, 25, 77, 79, 27, 83, 85, 29, 89, 91, 31, 95, 97, 33, 101, 103, 35, 107, 109, 37, 113, 115, 39, 119, 121, 41, 125, 127, 43, 131, 133, 45
Offset: 1

Views

Author

Artur Jasinski, Oct 31 2008

Keywords

Comments

From Jaroslav Krizek, May 28 2010: (Start)
a(n+1) = numerators of antiharmonic mean of the first n positive integers for n >= 1.
See A169609(n-1) - denominators of antiharmonic mean of the first n positive integers for n >= 1. (End)

Examples

			Fractions begin with 1/6, 1/2, 5/6, 7/6, 3/2, 11/6, 13/6, 5/2, 17/6, 19/6, 7/2, 23/6, ...

Links

Crossrefs

Cf. A146306, A146307, A146308, A146309, A146310, A146311, A146312, A146313.
Cf. A061347, A102283.
Trisections: A016921, A005408, A016969. [R. J. Mathar, Nov 21 2008]

Programs

  • Mathematica
    Table[Numerator[(2 n - 1)/6], {n, 1, 100}]
LinearRecurrence[{0,0,2,0,0,-1},{1,1,5,7,3,11},100] (* Harvey P. Dale, Feb 24 2015 *)
  • PARI
    a(n) = numerator((2*n-1)/3); \\ Altug Alkan, Apr 13 2018

Formula

From R. J. Mathar, Nov 21 2008: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x(1+x)(1+5x^2+x^4)/((1-x)^2*(1+x+x^2)^2). (End)
Sum_{k=1..n} a(k) ~ (7/9) * n^2. - Amiram Eldar, Apr 04 2024
a(n) = (2*n - 1)*(7 - A061347(n) +3*A102283(n))/9. - Stefano Spezia, Feb 14 2025

Extensions

Name edited by Altug Alkan, Apr 13 2018

A146307 a(n) = denominator of (n-6)/(2n) = denominator of (n+6)/(2n).

Original entry on oeis.org

2, 1, 2, 4, 10, 1, 14, 8, 6, 5, 22, 4, 26, 7, 10, 16, 34, 3, 38, 20, 14, 11, 46, 8, 50, 13, 18, 28, 58, 5, 62, 32, 22, 17, 70, 12, 74, 19, 26, 40, 82, 7, 86, 44, 30, 23, 94, 16, 98, 25, 34, 52, 106, 9, 110, 56, 38, 29, 118, 20, 122, 31, 42, 64, 130, 11, 134, 68, 46, 35, 142, 24
Offset: 1

Views

Author

Artur Jasinski, Oct 29 2008

Keywords

Comments

For numerators see A146306.
General formula:
2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) =
Hypergeometric2F1(A146306(n)/a(n), A146306(n+12)/a(n), 1/2, 3/4).
2*cos(2*Pi/n) is a root of a polynomial of degree EulerPhi(n)/2 = A000010(n)/2 = A023022(n).
Records in this sequence are even and are congruent to 2 or 10 mod 12 (see A091999).
Indices where odd numbers occur in this sequence are 4n-2 (see A016825).
Indices where prime numbers occur in this sequence see A146309.
From Robert Israel, Apr 21 2021: (Start)
a(n) = 2*n if n == 1, 5, 7 or 11 (mod 12).
a(n) = n if n == 4 or 8 (mod 12).
a(n) = 2*n/3 if n == 3 or 9 (mod 12).
a(n) = n/2 if n == 2 or 10 (mod 12).
a(n) = n/3 if n == 0 (mod 12).
a(n) = n/6 if n == 6 (mod 12). (End)
Sum_{k=1..n} a(k) ~ (77/144) * n^2. - Amiram Eldar, Apr 04 2024

Links

Crossrefs

Cf. A007310, A051724, A091999, A146306 (numerators), A146308.

Programs

  • Maple
    f:= n -> denom((n-6)/(2*n)):
map(f, [$1..100]); # Robert Israel, Apr 20 2021
  • Mathematica
    Table[Denominator[(n - 6)/(2 n)], {n, 1, 100}]
LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1},{2,1,2,4,10,1,14,8,6,5,22,4,26,7,10,16,34,3,38,20,14,11,46,8},80] (* Harvey P. Dale, May 15 2022 *)

A146308 a(n) is the smallest k such that the numerator of (k-6)/(2k) equals n.

Original entry on oeis.org

6, 7, 14, 15, 22, 11, 78, 13, 38, 33, 46, 17, 150, 19, 62, 51, 70, 23, 222, 25, 86, 69, 94, 29, 294, 31, 110, 87, 118, 35, 366, 37, 134, 105, 142, 41, 438, 43, 158, 123, 166, 47, 510, 49, 182, 141, 190, 53, 582, 55, 206, 159, 214, 59, 654, 61, 230, 177, 238, 65, 726
Offset: 0

Views

Author

Artur Jasinski, Oct 29 2008

Keywords

Comments

a(n) = index of first occurrence n in A146306.
General formula:
2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) = Hypergeometric2F1(A146306(n)/A146307(n), A146306(n+12)/A146307(n), 1/2, 3/4).
2*cos(2*Pi/n) is root of polynomial of degree = EulerPhi(n)/2 = A000010(n)/2 = A023022(n).

Links

Crossrefs

Cf. A007310, A051724, A146306, A146307, A146309.

Programs

  • Maple
    f:= proc(n) if n mod 6 = 0 then 12*n+6 elif n::even then 4*n+6 elif n mod 3 = 0 then 3*n+6 else n+6 fi end proc:
map(f, [$0..100]); # Robert Israel, Aug 05 2019
  • Mathematica
    aa = {}; Do[k = 1; While[Numerator[(k - 6)/(2 k)] != n, k++ ]; AppendTo[aa, k], {n, 0, 100}]; aa

Formula

From Robert Israel, Aug 05 2019: (Start)
If 6 | n then a(n) = 12*n+6
else if 3 | n then a(n) = 3*n+6
else if 2 | n then a(n) = 2*n+6
else a(n) = n+6.
a(n) = 2*a(n-6) - a(n-12).
G.f.: (6 + 7*x + 14*x^2 + 15*x^3 + 22*x^4 + 11*x^5 + 66*x^6 - x^7 + 10*x^8 + 3*x^9 + 2*x^10 - 5*x^11)/(1 - 2*x^6 + x^12). (End)
Showing 1-4 of 4 results.

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