A051543 -id:A051543 - cp's OEIS frontend

A025555 Least common multiple (or LCM) of first n positive triangular numbers (A000217).

1, 3, 6, 30, 30, 210, 420, 1260, 1260, 13860, 13860, 180180, 180180, 180180, 360360, 6126120, 6126120, 116396280, 116396280, 116396280, 116396280, 2677114440, 2677114440, 13385572200, 13385572200, 40156716600, 40156716600

Offset: 1

Clark Kimberling and Asher Auel

			a(5) = lcm{1, 3, 6, 10, 15} = 30.

T. D. Noe, Table of n, a(n) for n = 1..200
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.

Cf. A051543, A051538.

a025555 n = a025555_list !! (n-1)
a025555_list = scanl1 lcm $ tail a000217_list
-- Reinhard Zumkeller, Nov 22 2013

HalfFarey := proc (n) local a,b,c,d,k,s; if n<2 then RETURN([1]) fi; a:=0; b:=1; c:=1; d:=n; s:=NULL; do k := iquo(n+b,d); a,b,c,d := c, d, k*c-a, k*d-b; if b < 2*a then break fi; s := s, a/b od; [s] end:
A025555 := proc(n) local r; HalfFarey(n+1); subsop(nops(%) = NULL,%); mul(2*sin(Pi*r),r = %)^2 end: seq(round(evalf(A025555(i))),i=1..27); # Peter Luschny, Jun 09 2011

nn=30;With[{trnos=Accumulate[Range[nn]]},Table[LCM@@Take[trnos,n], {n,nn}]] (* Harvey P. Dale, Oct 21 2011 *)
f[x_] := x + 1; a[1] = f[1]; a[n_] := LCM[f[n], a[n - 1]]; Array[a, 30]/2 (* Robert G. Wilson v, Jan 04 2013 *)

S=1;for(n=1,20,S=lcm(S,n*(n+1)/2);print1(S,",")) \\ Edward Jiang, Sep 08 2014

a(n) = A003418(n+1)/2. - Matthew Vandermast, Jun 04 2012

Corrected by James Sellers

Definition rendered more precisely by Reinhard Zumkeller, Nov 22 2013

A265574 LCM-transform of triangular numbers.

Original entry on oeis.org

1, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 97, 1

Offset: 1

N. J. A. Sloane, Jan 02 2016

nonn

A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.

Cf. A051543, A020500, A014963.

LCMXfm:=proc(a) local L,i,n,g,b;
L:=nops(a);
g:=Array(1..L,0); b:=Array(1..L,0);
b[1]:=a[1]; g[1]:=a[1];
for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
lprint([seq(b[i],i=1..L)]);
end;
t1:=[seq(n*(n+1)/2,n=1..100)];
LCMXfm(t1);

LCMXfm[a_List] := Module[{L = Length[a], b, g}, b[1] = g[1] = a[[1]]; b[] = 0; g[] = 0; Do[g[n] = LCM[g[n - 1], a[[n]]]; b[n] = g[n]/g[n - 1], {n, 2, L}]; Array[b, L]];
LCMXfm[Table[n*(n + 1)/2, {n, 1, 100}]] (* Jean-François Alcover, Dec 05 2017, from Maple *)

From Andrey Zabolotskiy, Apr 11 2020: (Start)
a(n) = A051543(n-1) for n>1.
a(n) = A014963(n+1) for n>1. (End)

A051542 Quotients of consecutive values of LCM {b(1),...,b(n)}, b() = A000330.

Original entry on oeis.org

5, 14, 3, 11, 13, 2, 17, 19, 1, 23, 5, 3, 29, 62, 1, 1, 37, 1, 41, 43, 1, 47, 7, 1, 53, 1, 1, 59, 61, 2, 1, 67, 1, 71, 73, 1, 1, 79, 3, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 11, 1, 5, 254, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1

Offset: 1

Asher Auel

			a(3) = A051538(4)/A051538(3) = 210/70 = 3

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

Cf. A051538,A051543.

a051542 n = a051542_list !! (n-1)
a051542_list = zipWith div (tail a051538_list) a051538_list
-- Reinhard Zumkeller, Mar 12 2014

a(n) = A051538(n+1)/A051538(n)

Corrected and extended by James Sellers

Example fixed by Reinhard Zumkeller, Mar 12 2014

cp's OEIS Frontend

A025555 Least common multiple (or LCM) of first n positive triangular numbers (A000217).

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A265574 LCM-transform of triangular numbers.

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Maple

Mathematica

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A051542 Quotients of consecutive values of LCM {b(1),...,b(n)}, b() = A000330.

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Haskell

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