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.
a(1) = 1^(1*1-1) = 1 a(2) = 2^(2*2-2) = 4 a(3) = 3^(3*3-3) = 729 a(4) = 4^(2*4-4) = 256 a(5) = 5^(5*5-5) = 95367431640625 a(6) = 6^(1*1-6) = 0
A100994 := proc(n) if nops(numtheory[factorset](n)) <> 1 then 1 ; else n ; fi ; end: A014963 := proc(n) if nops(numtheory[factorset](n)) <> 1 then 1 ; else op(1,op(1,ifactors(n)[2])) ; fi ; end: A134579 := proc(n) local e ; e := A014963(n)*A100994(n)-n ; if e >= 0 then n^e ; else 0 ; fi ; end: seq(A134579(n),n=1..13) ; # R. J. Mathar, Jan 30 2008
LCM of {1,2,3,4,5,6} = 60. The primes up to 6 are 2, 3 and 5. floor(log(6)/log(2)) = 2 so the exponent of 2 is 2.
floor(log(6)/log(3)) = 1 so the exponent of 3 is 1.
floor(log(6)/log(5)) = 1 so the exponent of 5 is 1. Therefore, a(6) = 2^2 * 3^1 * 5^1 = 60. - _David A. Corneth_, Jun 02 2017
a003418 = foldl lcm 1 . enumFromTo 2 -- Reinhard Zumkeller, Apr 04 2012, Apr 25 2011
[1] cat [Exponent(SymmetricGroup(n)) : n in [1..28]]; // Arkadiusz Wesolowski, Sep 10 2013
[Lcm([1..n]): n in [0..30]]; // Bruno Berselli, Feb 06 2015
A003418 := n-> lcm(seq(i,i=1..n)); HalfFarey := proc(n) local a,b,c,d,k,s; 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 2*a > b then break fi; s := s,(a/b); od: [s] end: LCM := proc(n) local i; (1/2)*mul(2*sin(Pi*i),i=HalfFarey(n))^2 end: # Peter Luschny # next Maple program: a:= proc(n) option remember; `if`(n=0, 1, ilcm(n, a(n-1))) end: seq(a(n), n=0..33); # Alois P. Heinz, Jun 10 2021
Table[LCM @@ Range[n], {n, 1, 40}] (* Stefan Steinerberger, Apr 01 2006 *)
FoldList[ LCM, 1, Range@ 28]
A003418[0] := 1; A003418[1] := 1; A003418[n_] := A003418[n] = LCM[n,A003418[n-1]]; (* Enrique Pérez Herrero, Jan 08 2011 *)
Table[Product[Prime[i]^Floor[Log[Prime[i], n]], {i, PrimePi[n]}], {n, 0, 28}] (* Wei Zhou, Jun 25 2011 *)
Table[Product[Cyclotomic[n, 1], {n, 2, m}], {m, 0, 28}] (* Fred Daniel Kline, May 22 2014 *)
a1[n_] := 1/12 (Pi^2+3(-1)^n (PolyGamma[1,1+n/2] - PolyGamma[1,(1+n)/2])) // Simplify
a[n_] := Denominator[Sqrt[a1[n]]];
Table[If[IntegerQ[a[n]], a[n], a[n]*(a[n])[[2]]], {n, 0, 28}] (* Gerry Martens, Apr 07 2018 [Corrected by Vaclav Kotesovec, Jul 16 2021] *)
a(n)=local(t); t=n>=0; forprime(p=2,n,t*=p^(log(n)\log(p))); t
a(n)=if(n<1,n==0,1/content(vector(n,k,1/k)))
a(n)=my(v=primes(primepi(n)),k=sqrtint(n),L=log(n+.5));prod(i=1,#v,if(v[i]>k,v[i],v[i]^(L\log(v[i])))) \\ Charles R Greathouse IV, Dec 21 2011
a(n)=lcm(vector(n,i,i)) \\ Bill Allombert, Apr 18 2012 [via Charles R Greathouse IV]
n=1; lim=100; i=1; j=1; until(n==lim, a=lcm(j,i+1); i++; j=a; n++; print(n" "a);); \\ Mike Winkler, Sep 07 2013
from functools import reduce
from operator import mul
from sympy import sieve
def integerlog(n,b): # find largest integer k>=0 such that b^k <= n
kmin, kmax = 0,1
while b**kmax <= n:
kmax *= 2
while True:
kmid = (kmax+kmin)//2
if b**kmid > n:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
return kmin
def A003418(n):
return reduce(mul,(p**integerlog(n,p) for p in sieve.primerange(1,n+1)),1) # Chai Wah Wu, Mar 13 2021
# generates initial segment of sequence from math import gcd from itertools import accumulate def lcm(a, b): return a * b // gcd(a, b) def aupton(nn): return [1] + list(accumulate(range(1, nn+1), lcm)) print(aupton(30)) # Michael S. Branicky, Jun 10 2021
[lcm(range(1,n)) for n in range(1, 30)] # Zerinvary Lajos, Jun 06 2009
(define (A003418 n) (let loop ((n n) (m 1)) (if (zero? n) m (loop (- n 1) (lcm m n))))) ;; Antti Karttunen, Jan 03 2018
Concatenation([1],List([1..70],n->1-Lcm(List([1..n+1],i->i))/Lcm(List([1..n],i->i)))); # Muniru A Asiru, Mar 04 2019
A120112:= func< n | n eq 0 select 1 else 1-Lcm([1..n+1])/Lcm([1..n]) >; [A120112(n): n in [0..100]]; // G. C. Greubel, May 05 2023
Table[If[n==0, 1, 1-LCM@@Range[n+1]/(LCM@@Range[n])], {n,0,100}] (* G. C. Greubel, May 05 2023 *)
a(n) = if (n==0, 1, 1 - lcm(vector(n+1, k, k))/lcm(vector(n, k, k))); \\ Michel Marcus, Sep 11 2016
def A120112(n): return 1 if n == 0 else 1 - lcm(range(1,n+2)) // lcm(range(1,n+1)) [A120112(n) for n in range(101)] # G. C. Greubel, May 05 2023
b:= proc(n) b(n):= `if`(n=0, 1, ilcm(n, b(n-1))) end: a:= proc(n) a(n):= `if`(n<0, 0, a(n-1) +b(n)) end: seq(a(n), n=0..35); # Alois P. Heinz, Mar 31 2018
Table[If[n == 0, 1, LCM @@ Range[n]], {n, 0, 50}] // Accumulate (* Jean-François Alcover, Jan 03 2022 *)
a(n) = sum(k=0, n, lcm(vector(k, i, i))); \\ Michel Marcus, Mar 13 2018
(define (A173185 n) (if (< n 1) 1 (+ (A173185 (- n 1)) (A003418 n))))
The least common multiple of the first few rows are:
lcm{1} = 1
lcm{1,2} = 2
lcm{1,2,3} = 6
lcm{1,1,3,4} = 12
lcm{1,1,3,4,5} = 60
lcm{1,1,3,4,5,7} = 420
lcm{1,1,3,1,5,7,8} = 840
lcm{1,1,1,1,5,7,8,9} = 2520
lcm{1,1,1,1,5,7,8,9,11} = 27720
Multiplying the terms in the rows produces the same result:
1 = 1
1*2 = 2
1*2*3 = 6
1*1*3*4 = 12
1*1*3*4*5 = 60
1*1*3*4*5*7 = 420
1*1*3*1*5*7*8 = 840
1*1*1*1*5*7*8*9 = 2520
1*1*1*1*5*7*8*9*11 = 27720
a(9) = 27 = 81/3 where 9^2 = 81 and A048671(9) = 3. a(9) = 27 = 9*A014963(n) = 9*3.
lcm{1}= 1
lcm{1,1} = 1
lcm{1,1,2} = 2
lcm{1,1,2,3} = 6
lcm{1,1,4,3,1} = 12
lcm{1,1,4,3,1,5} = 60
lcm{1,1,4,3,1,5,1} = 60
lcm{1,1,4,3,1,5,1,7} = 420
lcm{1,1,8,3,1,5,1,7,1} = 840
lcm{1,1,8,9,1,5,1,7,1,1} = 2520
1 = 1
1*1 = 1
1*1*2 = 2
1*1*2*3 = 6
1*1*4*3*1 = 12
1*1*4*3*1*5 = 60
1*1*4*3*1*5*1 = 60
1*1*4*3*1*5*1*7 = 420
1*1*8*3*1*5*1*7*1 = 840
1*1*8*9*1*5*1*7*1*1 = 2520
=if(column()=1;1; if(and(row()-1>=(A089026(n-1))^0;row()-1<(A089026(n-1))^1);(A089026(n-1))^0; if(and(row()-1>=(A089026(n-1))^1;row()-1<(A089026(n-1))^2);(A089026(n-1))^1; if(and(row()-1>=(A089026(n-1))^2;row()-1<(A089026(n-1))^3);(A089026(n-1))^2; if(and(row()-1>=(A089026(n-1))^3;row()-1<(A089026(n-1))^4);(A089026(n-1))^3; if(and(row()-1>=(A089026(n-1))^4;row()-1<(A089026(n-1))^5);(A089026(n-1))^4; 1)))))) And so on.
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