A033931 -id:A033931 - cp's OEIS frontend

A067046 a(n) = lcm(n, n+1, n+2)/6.

1, 2, 10, 10, 35, 28, 84, 60, 165, 110, 286, 182, 455, 280, 680, 408, 969, 570, 1330, 770, 1771, 1012, 2300, 1300, 2925, 1638, 3654, 2030, 4495, 2480, 5456, 2992, 6545, 3570, 7770, 4218, 9139, 4940, 10660, 5740, 12341, 6622, 14190, 7590, 16215, 8648, 18424, 9800

Offset: 1

Amarnath Murthy, Dec 30 2001

			a(6) = 28 as lcm(6,7,8)/6 = 168/6 = 28.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Amarnath Murthy, Some Notions on Least Common Multiples, Smarandache Notions Journal, Vol. 12, No. 1-2-3 (Spring 2001), pp. 307-308.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).
Index entries for sequences related to lcm's.

Cf. A000447 (bisection), A006331 (bisection), A033931.

a067046 = (`div` 6) . a033931  -- Reinhard Zumkeller, Jul 04 2012

Table[LCM[n,n+1,n+2]/6,{n,50}] (* Harvey P. Dale, Jan 11 2011 *)

a(n)={lcm([n, n+1, n+2])/6} \\ Harry J. Smith, Apr 30 2010

a(n)=binomial(n+2,3)/(2-n%2) \\ Charles R Greathouse IV, Feb 27 2012

G.f.: (x^4 + 2x^3 + 6x^2 + 2x + 1)/(1 - x^2)^4.
a(n) = binomial(n+2,3)*(3-(-1)^n)/4. - Gary Detlefs, Apr 13 2011
Quasipolynomial: a(n) = n(n+1)(n+2)/6 when n is odd and n(n+1)(n+2)/12 otherwise. - Charles R Greathouse IV, Feb 27 2012
a(n) = A033931(n) / 6. - Reinhard Zumkeller, Jul 04 2012
From Amiram Eldar, Sep 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*(1 - log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(3*log(2) - 2). (End)

A188386 a(n) = numerator(H(n+2)-H(n-1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

Original entry on oeis.org

11, 13, 47, 37, 107, 73, 191, 121, 299, 181, 431, 253, 587, 337, 767, 433, 971, 541, 1199, 661, 1451, 793, 1727, 937, 2027, 1093, 2351, 1261, 2699, 1441, 3071, 1633, 3467, 1837, 3887, 2053, 4331, 2281, 4799, 2521, 5291, 2773, 5807, 3037, 6347, 3313, 6911, 3601

Offset: 1

Gary Detlefs, Mar 29 2011

Denominators are listed in A033931.
A027446 appears to be divisible by a(n).
The sequence lists also the largest odd divisors of 3*m^2-1 (A080663) for m>1. In fact, for m even, the largest odd divisor is 3*m^2-1 itself; for m odd, the largest odd divisor is (3*m^2-1)/2. From this follows the second formula given in Formula field. - Bruno Berselli, Aug 27 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A033931 (denominators), A001008, A001711, A027446, A002805, A003154, A080663, A158463, A213998.

import Data.Ratio ((%), numerator)
a188386 n = a188386_list !! (n-1)
a188386_list = map numerator $ zipWith (-) (drop 3 hs) hs
   where hs = 0 : scanl1 (+) (map (1 %) [1..])
-- Reinhard Zumkeller, Jul 03 2012

[Numerator((3*n^2+6*n+2)/((n*(n+1)*(n+2)))): n in [1..50]]; // Vincenzo Librandi, Mar 30 2011

seq((3-(-1)^n)*(3*n^2+6*n+2)/4, n=1..100);

Table[(3 - (-1)^n)*(3*n^2 + 6*n + 2)/4, {n, 40}] (* Wesley Ivan Hurt, Jan 29 2017 *)
Numerator[#[[4]]-#[[1]]]&/@Partition[HarmonicNumber[Range[0,50]],4,1] (* or *) LinearRecurrence[{0,3,0,-3,0,1},{11,13,47,37,107,73},50] (* Harvey P. Dale, Dec 31 2017 *)

a(n) = numerator((3*n^2+6*n+2)/(n*(n+1)*(n+2))).
a(n) = (3-(-1)^n)*(3*n^2+6*n+2)/4.
a(2n+1) = A158463(n+1), a(2n) = A003154(n+1).
G.f.: -x*(11+13*x+14*x^2-2*x^3-x^4+x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Apr 09 2011
a(n) = numerator of coefficient of x^3 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - Francesco Daddi, Aug 04 2011
H(n+3) = 3/2 + 2*f(n)/((n+2)*(n+3)), where f(n) = Sum_{k=0..n}((-1)^k*binomial(-3,k)/(n+1-k)). - Gary Detlefs, Jul 17 2011
a(n) = A213998(n+2,2). - Reinhard Zumkeller, Jul 03 2012
Sum_{n>=1} 1/a(n) = c*(tan(c) - cot(c)/2) - 1/2, where c = Pi/(2*sqrt(3)). - Amiram Eldar, Sep 27 2022

A078637 a(n) = rad(n(n+1)(n+2)), where rad(m) is the largest squarefree number dividing m (see A007947).

Original entry on oeis.org

6, 6, 30, 30, 210, 42, 42, 30, 330, 330, 858, 546, 2730, 210, 510, 102, 1938, 570, 3990, 2310, 10626, 1518, 690, 390, 390, 546, 1218, 6090, 26970, 930, 2046, 1122, 39270, 3570, 7770, 4218, 54834, 7410, 15990, 8610, 74046, 19866, 14190, 7590, 32430, 6486

Offset: 1

Jon Perry, Dec 12 2002

nonn

			a(3) = rad(3*4*5) = 30.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007531, A007947, A033931.

a078637 n = a007947 $ product [n..n+2] -- Reinhard Zumkeller, Jul 04 2012

with(numtheory):rad:=proc(n) local s,i: s:=ifactors(n)[2]: RETURN(mul(s[i][1],i=1..nops(s))): end; seq(rad(n*(n+1)*(n+2)),n=1..60); seq(piecewise(n mod 2=0,rad(n/2)*rad(n+1)*rad(n/2+1),rad(n)*rad(n+1)*rad(n+2)),n=1..60); (C. Ronaldo)

lsf[n_]:=Max[Select[Divisors[n],SquareFreeQ]]; lsf/@Table[n(n+1)(n+2),{n,50}] (* Harvey P. Dale, Oct 18 2020 *)
a[n_] := Times @@ Union @@ (FactorInteger[#][[;; , 1]] & /@ (n + {0, 1, 2})); Array[a, 50] (* Amiram Eldar, Jun 30 2022 *)

rad(n)=local(p,i); p=factor(n)[,1]; prod(i=1,length(p),p[i])
for (k=1,100,print1(rad(k*(k+1)*(k+2))","))

a(n) = rad(n)*rad(n+1)*rad(n+2) if n is odd; or rad(n/2)*rad(n+1)*rad(n/2+1) if n is even. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 13 2004
a(n) = A007947(A033931(n)). - Reinhard Zumkeller, Jul 04 2012
a(n) = A007947(A007531(n+2)). - Amiram Eldar, Jun 30 2022

A364004 Orders of simple groups PSL(2,K) with exactly 4 prime divisors.

Original entry on oeis.org

660, 1092, 4080, 3420, 6072, 7800, 9828, 14880, 32736, 25308, 51888, 58800, 74412, 194472, 265680, 456288, 612468, 1024128, 2097024, 2165292, 3594432, 7174332, 8487168, 28090752, 57750408, 96049728, 321367392

Offset: 1

Lixin Zheng, Jul 03 2023

nonn

Sequence is conjectured to be infinite, see Bugeaud et al.

All entries are divisible by 6 by order formula for PSL(2,q).

			660 has prime divisors 2,3,5,11.

Y. Bugeaud, Z. Cao, and M. Mignotte, On Simple K4-Groups, Journal of Algebra, 241 (2001), 658-668.

Subsequence of A352806. Elements generated from A364003.

Terms are q*(q^2-1)/gcd(2, q-1) for q in A364003.

a(n) = A033931(A364003(n)-1).

cp's OEIS Frontend

A067046 a(n) = lcm(n, n+1, n+2)/6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

A188386 a(n) = numerator(H(n+2)-H(n-1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

A078637 a(n) = rad(n(n+1)(n+2)), where rad(m) is the largest squarefree number dividing m (see A007947).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A364004 Orders of simple groups PSL(2,K) with exactly 4 prime divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula