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.
The matching fractions are 1/2, 1/3, 2/5, 9/23, 30/73, 59/143 ... (this is A182975/A182976).
Right side of Stern-Brocot tree: 1/1 2/1 3/2 3/1 4/3 5/3 5/2 4/1 5/4 7/5 8/5 7/4 7/3 8/3 7/2 5/1 A038567/A038566: 1/1 2/1 3/1 3/2 4/1 4/3 5/1 5/2 5/3 5/4 6/1 6/5 7/1 7/2 7/3 7/4
a(1)=1, a(2)=2 and a(3)=7 because the 1st, 2nd and 7th fractions match in the following two sequences: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5 ... (A020652/A038567) 1/2, 1/3, 1/4, 2/3, 1/5, 1/6, 2/5, 3/4 ... (A182972/A182973)
The matching fractions are 1/2, 1/3, 2/5, 9/23, 30/73, 59/143 ... (this is A182975/A182976).
Row 1: 1/1 5 Row 2: 1/2 11 Row 3: 1/3 2/3 29 37 Row 4: 1/4 3/4 6421 367 Row 5: 1/5 2/5 3/5 4/5 149 14281 251 Row 6: 1/6 5/6 521 631 Row 7: 1/7 .. 6/7 84913 127 331 75479 787 7057 Row 8: 1/8 3/8 5/8 7/8 1949 3407 388621 1847 etc.
f[n_] := Block[{p = 2, q = 3, r = 5}, While[(r - q) != n(q - p), p = q; q = r; r = NextPrime@ r]; q]; Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}]; ff = Rest@ Reverse@ Sort[ Farey[25], Denominator[#2] < Denominator[#1] &]; f@# & /@ ff
Row 1: 1/1 5 Row 2: 1/2 3 Row 3: 1/3 2/3 31 23 Row 4: 1/4 3/4 8123 89 Row 5: 1/5 2/5 3/5 4/5 139 7963 337 409 Row 6: 1/6 5/6 199 797 Row 7: 1/7 .. 6/7 45439 113 953 88547 293 2633 Row 8: 1/8 3/8 5/8 7/8 1933 3643 137029 13381 etc.
f[n_] := Block[{p = 2, q = 3, r = 5}, While[q != n(r - q) + p, p = q; q = r; r = NextPrime@ r]; q]; Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}]; ff = Rest@ Reverse@ Sort[ Farey[25], Denominator[#2] < Denominator[#1] &]; f@# & /@ ff
a(5) = 19 == A020652(5) = 3 (mod A038567(5) = 4) and is the least prime > a(4) = 13 with this property.
N:= 100: # for a(1)..a(N)
A:=Vector(N): A[1]:= 3: n:= 1:
for d from 3 while n < N do
for m from 1 to d-1 while n < N do
if igcd(m,d)=1 then
n:= n+1;
for k from ceil((A[n-1]+1 - m)/d) do
q:= d*k+m;
if isprime(q) then A[n]:= q; break fi
od
fi
od od:
convert(A,list);
a(n = 1) = 23 because A038566(23)/A038567(22) = 1/9, the unique fraction (in lowest terms) whose decimal expansion is 0.111..., i.e., period = (1), repeated. a(n = 2) = 24 because A038566(24)/A038567(23) = 2/9, the unique fraction (in lowest terms) whose decimal expansion is 0.222..., i.e., period = (2), repeated. a(n = 3) = 3 because A038566(3)/A038567(2) = 1/3, the unique fraction (in lowest terms) whose decimal expansion is 0.333..., i.e., period = (3), repeated. a(n = 9) = 1 because A038566(1)/A038567(0) = 1/1, the unique fraction (in lowest terms) equal to 0.999..., i.e., period = (9), repeated. a(n = 10) = 2951 because A038566(2951)/A038567(2950) = 10/99, the unique fraction (in lowest terms) whose decimal expansion is 0.1010..., i.e., period = (10), repeated. a(11) = a(1) because the unique fraction that has decimal expansion 0.1111..., i.e., period (11) repeated, is 1/9, the same as for 0.111..., i.e., period (1), repeated.
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
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