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 sequence starts with 1,1,2,1,3,1,3,2,2,2,2,2,3,... The product a(n)*a(n+1) = 1 is true exactly once [this is the product a(1) * a(2) = 1 * 1 = 1]; The product a(n)*a(n+1) = 2 is true exactly twice [these are the products a(2) * a(3) = 1 * 2 = 2 and a(3) * a(4) = 2 * 1 = 2]; The product a(n)*a(n+1) = 3 is true exactly three times [these are the products a(4) * a(5) = 1 * 3 = 3 ; a(5) * a(6) = 3 * 1 = 3, and a(6) * a(7) = 1 * 3 = 3]; ... The product a(n)*a(n+1) = 4 is true exactly four times [these are the products a(8) * a(9) = 2 * 2 = 4 ; a(9) * a(10) = 2 * 2 = 4 ; a(10) * a(11) = 2 * 2 = 4 ; a(11) * a(12) = 2 * 2 = 4] ; and so on.
nmax = 1000; time = {0}; v = 1;
A307720 = Reap[For[n = 1, n <= nmax, n++, Sow[v]; For[o = 1, True, o++, While[Length[time] < o*v, time = Join[time, Table[0, {Length[time]}]]]; If[time[[o*v]]+1 <= o*v, time[[o*v]]++; v = o; Break[]]]]][[2, 1]] (* Jean-François Alcover, Oct 23 2021, after Rémy Sigrist's PARI program *)
\\ See Links section.
from itertools import islice from collections import Counter def A307720(): # generator of terms. Greedy algorithm yield 1 c, b = Counter(), 1 while True: k, kb = 1, b while c[kb] >= kb: k += 1 kb += b c[kb] += 1 b = k yield k A307720_list = list(islice(A307720(),100)) # Chai Wah Wu, Oct 21 2021
Given that the sequence begins 1,1,2,2,... then a(5)=3, since either of the choices a(5)=1 or a(5)=2 would lead to a repetition of one of the previous products 1,2,4 of adjacent pairs of terms.
A[1]:= 1: A[2]:= 1: S:= {1}:
for n from 3 to 100 do
Sp:= select(type,map(s -> s/A[n-1],S),integer);
if nops(Sp) = Sp[-1] then A[n]:= Sp[-1]+1
else A[n]:= min({$1..Sp[-1]} minus Sp)
fi;
S:= S union {A[n-1]*A[n]};
od:
seq(A[n],n=1..100); # Robert Israel, Aug 28 2014
t = {1, 1}; Do[AppendTo[t, 1]; While[Length[Union[Most[t]*Rest[t]]] < n - 1, t[[-1]]++], {n, 3, 100}]; t (* T. D. Noe, Nov 16 2011 *)
from itertools import islice def A088177(): # generator of terms yield 1 yield 1 p, a = {1}, 1 while True: n = 1 while n*a in p: n += 1 p.add(n*a) a = n yield n A088177_list = list(islice(A088177(),20)) # Chai Wah Wu, Oct 21 2021
// See Links section.
See Links section.
from itertools import islice from collections import Counter def A348446(): # generator of terms. Greedy algorithm a = 1 c, b = Counter(), 1 while True: k, kb = 1, b while c[kb] >= kb: k += 1 kb += b c[kb] += 1 b = k a2 = k yield a-a2 k, kb = 1, b while c[kb] >= kb: k += 1 kb += b c[kb] += 1 b = k a = k A348446_list = list(islice(A348446(),100)) # Chai Wah Wu, Oct 23 2021
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