A051724 -id:A051724 - cp's OEIS frontend

A109053 a(n) = lcm(n,12).

0, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 132, 12, 156, 84, 60, 48, 204, 36, 228, 60, 84, 132, 276, 24, 300, 156, 108, 84, 348, 60, 372, 96, 132, 204, 420, 36, 444, 228, 156, 120, 492, 84, 516, 132, 180, 276, 564, 48, 588, 300, 204, 156, 636, 108, 660, 168

Offset: 0

Mitch Harris, Jun 18 2005

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A051724, A109042.

List([0..60],n->Lcm(n,12)); # Muniru A Asiru, Mar 04 2019

[LCM(n, 12): n in [0..60]]; // G. C. Greubel, Mar 06 2019

Array[LCM[#,12]&,60,0] (* Harvey P. Dale, Mar 26 2015 *)

concat(0, Vec(12*x*(1 + x + x^2 + x^3 + 5*x^4 + x^5 + 7*x^6 + 2*x^7 + 3*x^8 + 5*x^9 + 11*x^10 + x^11 + 11*x^12 + 5*x^13 + 3*x^14 + 2*x^15 + 7*x^16 + x^17 + 5*x^18 + x^19 + x^20 + x^21 + x^22) / (1 - 2*x^12 + x^24) + O(x^40))) \\ Colin Barker, Mar 04 2019

for(n=0,60, print1(lcm(n,12), ", ")) \\ G. C. Greubel, Mar 06 2019

[lcm(n,12) for n in range(0,57)] # Zerinvary Lajos, Jun 09 2009

a(n) = n*12/gcd(n, 12).
a(n) = 12*A051724(n). - R. J. Mathar, Feb 12 2019
From Colin Barker, Mar 04 2019: (Start)
G.f.: 12*x*(1 + x + x^2 + x^3 + 5*x^4 + x^5 + 7*x^6 + 2*x^7 + 3*x^8 + 5*x^9 + 11*x^10 + x^11 + 11*x^12 + 5*x^13 + 3*x^14 + 2*x^15 + 7*x^16 + x^17 + 5*x^18 + x^19 + x^20 + x^21 + x^22) / (1 - 2*x^12 + x^24).
a(n) = 2*a(n-12) - a(n-24) for n>23.
(End)
Sum_{k=1..n} a(k) ~ (77/24) * n^2. - Amiram Eldar, Nov 26 2022

A146306 a(n) = numerator of (n-6)/(2n).

Original entry on oeis.org

-5, -1, -1, -1, -1, 0, 1, 1, 1, 1, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 4, 17, 3, 19, 5, 7, 11, 23, 2, 25, 13, 9, 7, 29, 5, 31, 8, 11, 17, 35, 3, 37, 19, 13, 10, 41, 7, 43, 11, 15, 23, 47, 4, 49, 25, 17, 13, 53, 9, 55, 14, 19, 29, 59, 5, 61, 31, 21, 16, 65, 11, 67, 17, 23, 35, 71, 6, 73, 37

Offset: 1

Artur Jasinski, Oct 29 2008

For denominators see A146307.
General formula:
2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) =
Hypergeometric2F1(a(n)/A146307(n), a(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).
Records in this sequence are congruent to 1 or 5 mod 6 (see A007310).
First occurrence n in this sequence see A146308.

			Fractions begin with -5/2, -1, -1/2, -1/4, -1/10, 0, 1/14, 1/8, 1/6, 1/5, 5/22, 1/4, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A000010, A007310, A023022, A051724, A146307 (denominators), A146308.

Table[Numerator[(n - 6)/(2 n)], {n, 1, 100}]

a(n+5) = A051724(n).
Sum_{k=1..n} a(k) ~ (77/288) * n^2. - Amiram Eldar, Apr 04 2024
From Chai Wah Wu, May 08 2025: (Start)
a(n) = 2*a(n-12) - a(n-24) for n > 24.
G.f.: x*(x^23 + 7*x^22 + 2*x^21 + 3*x^20 + 5*x^19 + 11*x^18 + x^17 + 13*x^16 + 7*x^15 + 5*x^14 + 4*x^13 + 17*x^12 + x^11 + 5*x^10 + x^9 + x^8 + x^7 + x^6 - x^4 - x^3 - x^2 - x - 5)/(x^24 - 2*x^12 + 1). (End)

A106616 Numerator of n/(n+15).

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 2, 7, 8, 3, 2, 11, 4, 13, 14, 1, 16, 17, 6, 19, 4, 7, 22, 23, 8, 5, 26, 9, 28, 29, 2, 31, 32, 11, 34, 7, 12, 37, 38, 13, 8, 41, 14, 43, 44, 3, 46, 47, 16, 49, 10, 17, 52, 53, 18, 11, 56, 19, 58, 59, 4, 61, 62, 21, 64, 13, 22, 67, 68, 23, 14, 71, 24, 73, 74, 5, 76, 77, 26

Offset: 0

N. J. A. Sloane, May 15 2005

Multiplicative and also a strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Feb 24 2019

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A000010, A106614, A106615, A106617, A106618, A106619, A106620, A106621.

Cf. Other sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([0..80],n->NumeratorRat(n/(n+15))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+15)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+15)),n=0..100); # Nathaniel Johnston, Apr 18 2011

f[n_]:=Numerator[n/(n+15)]; Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011*)

a(n) = numerator(n/(n+15)); \\ Michel Marcus, Feb 19 2017

[lcm(n,15)/15 for n in range(0, 79)] # Zerinvary Lajos, Jun 09 2009

Dirichlet g.f.: zeta(s-1)*(1-4/5^s-2/3^s+8/15^s). - R. J. Mathar, Apr 18 2011
a(n) = gcd((n-2)*(n-1)*n*(n+1)*(n+2)/15, n) for n>=1. - Lechoslaw Ratajczak, Feb 19 2017
From Peter Bala, Feb 24 2019: (Start)
a(n) = n/gcd(n,15), a quasi-polynomial in n since gcd(n,15) is a purely periodic sequence of period 15.
O.g.f.: F(x) - 2*F(x^3) - 4*F(x^5) + 8*F(x^15), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 15} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (2/3)*log(1/(1 - x^3)) + (4/5)*log(1/(1 - x^5)) + (8/15)*log(1/(1 - x^15)), where phi(n) denotes the Euler totient function A000010. (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(3^e) = 3^max(0,e-1), a(5^e) = 5^max(0,e-1), and a(p^e) = p^e otherwise.
Sum_{k=1..n} a(k) ~ (49/150) * n^2. (End)

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

Artur Jasinski, Oct 29 2008

nonn

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).

A007310, A051724, A146306, A146307, A146308.

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

A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 1, 0, -1, -1, 1, 3, 1, -1, -3, -1, 1, 2, 1, 0, -1, -2, -1, 1, 5, 3, 1, -1, -3, -5, -1, 1, 3, 1, 1, 0, -1, -1, -3, -1, 1, 7, 5, 1, 1, -1, -1, -5, -7, -1, 1, 4, 3, 2, 1, 0, -1, -2, -3, -4, -1, 1, 0, 7, 5, 3, 1, -1, -3, -5, -7, -9, -1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367727.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the numerators begins:
  1, -1, -1, -1, -1, -1, -1, -1, ...
  1,  0, -1, -1, -3, -2, -5, -3, ...
  1,  1,  0, -1, -1, -3, -1, -5, ...
  1,  1,  1,  0, -1, -1, -1, -2, ...
  1,  3,  1,  1,  0, -1, -1, -3, ...
  1,  2,  3,  1,  1,  0, -1, -1, ...
  1,  5,  1,  1,  1,  1,  0, -1, ...
  1,  3,  5,  2,  3,  1,  1,  0, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367825 (denominator), A367826 (antidiagonal sums).

Cf. A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106617, A106619, A153881 (n=0), A231190, A367727 (k=1).

A[0,0]=1; A[n_,k_]:=Numerator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(1, n) = -A026741(n-1) for n > 0.
A(2, n) = -A060819(n-2) for n > 2.
A(3, n) = -A060789(n-3) for n > 3.
A(4, n) = -A106609(n-4) for n > 3.
A(5, n) = -A106611(n-5) for n > 4.
A(6, n) = -A051724(n-6) for n > 5.
A(7, n) = -A106615(n-7) for n > 6.
A(8, n) = -A106617(n-8) = A231190(n) for n > 7.
A(9, n) = -A106619(n-9) for n > 8.
A(10, n) = -A106612(n-10) for n > 9.

A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 1, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 3, 1, 3, 3, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 4, 2, 4, 1, 4, 2, 4, 1, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 4, 6, 12, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367728.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the denominators begins:
  1, 1, 1, 1,  1,  1,  1,  1, ...
  1, 1, 3, 2,  5,  3,  7,  4, ...
  1, 3, 1, 5,  3,  7,  2,  9, ...
  1, 2, 5, 1,  7,  4,  3,  5, ...
  1, 5, 3, 7,  1,  9,  5, 11, ...
  1, 3, 7, 4,  9,  1, 11,  6, ...
  1, 7, 2, 3,  5, 11,  1, 13, ...
  1, 4, 9, 5, 11,  6, 13,  1, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367824 (numerator), A367827 (antidiagonal sums).

Cf. A000012 (n=0), A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106615, A106617, A231190, A367728 (k=1).

A[0,0]=1; A[n_,k_]:=Denominator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(1, n) = A026741(n+1).
A(2, n) = A060819(n+2).
A(3, n) = A060789(n+3).
A(4, n) = A106609(n+4).
A(5, n) = A106611(n+5).
A(6, n) = A051724(n+6).
A(7, n) = A106615(n+7).
A(8, n) = A106617(n+8) = A231190(n+16).
A(9, n) = A106619(n+9).
A(10, n) = A106612(n+10).

cp's OEIS Frontend

A109053 a(n) = lcm(n,12).

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A146306 a(n) = numerator of (n-6)/(2n).

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A106616 Numerator of n/(n+15).

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

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A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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