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.
A088178(5) = 3, so a(3) = 5.
from itertools import islice def A348442(): # generator of terms yield 1 c, p, a = 1, {1}, 1 while True: n, na = 1, a while na in p: n += 1 na += a p.add(na) a = n if c < na: c = na yield c A348442_list = list(islice(A348442(),100)) # Chai Wah Wu, Oct 21 2021
from itertools import islice def A348443(): # generator of terms yield 1 c, p, a, i = 1, {1}, 1, 1 while True: n, na = 1, a while na in p: n += 1 na += a p.add(na) a = n i += 1 if c < na: c = na yield i A348443_list = list(islice(A348443(),100)) # Chai Wah Wu, Oct 21 2021
The first terms in this sequence and in A307720 are: n a(n) A307720(n) -- ---- ---------- 1 1 1 2 2 1 3 2 2 4 3 1 5 3 3 6 3 1 7 6 3 8 4 2 9 4 2 10 4 2
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from itertools import islice from collections import Counter def A307730(): # generator of terms. Greedy algorithm c, b = Counter(), 1 while True: k, kb = 1, b while c[kb] >= kb: k += 1 kb += b c[kb] += 1 b = k yield kb A307730_list = list(islice(A307730(),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
Block[{c, m = 1, n}, Union@ FoldList[Max, {1}~Join~Reap[Do[n = 1; While[IntegerQ[c[m n]], n++]; Sow[n]; Set[c[m n], 1]; m = n, 2^12]][[-1, -1]]]] (* Michael De Vlieger, Oct 21 2021 *)
from itertools import islice def A348440(): # generator of terms yield 1 c, p, a = 1, {1}, 1 while True: n, na = 1, a while na in p: n += 1 na += a p.add(na) a = n if c < n: c = n yield c A348440_list = list(islice(A348440(),100)) # Chai Wah Wu, Oct 21 2021
The first terms, alongside a(n)*a(n+1)^2, are: n a(n) a(n)*a(n+1)^2 -- ---- ------------- 1 1 1 2 1 4 3 2 2 4 1 9 5 3 3 6 1 16 7 4 36 8 3 12 9 2 8 10 2 18
s=0; v=1; for(n=1, 83, print1(v", "); for (w=1, oo, if (!bittest(s,x=v*w^2), s+=2^x; v=w; break)))
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