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.
Table showing n, A353709(n), and b(n), the binary expansion of A353709(n) writing "." for zeros for clarity. a(n) interprets 1's in the k-th place of b(n) as prime(k+1) and thereafter takes the product. We find a(n) = A005117(j). n A353709(n) b(n) a(n) j ---------------------------- 1 0 . 1 1 2 1 1 2 2 3 2 1. 3 3 4 4 1.. 5 4 5 8 1... 7 6 6 3 11 6 5 7 16 1.... 11 8 8 12 11.. 35 23 9 32 1..... 13 9 10 17 1...1 22 15 11 6 11. 15 11 12 40 1.1... 91 57 13 64 1...... 17 12 14 5 1.1 10 7 15 10 1.1. 21 14 16 48 11.... 143 89 ...
nn = 2^7; c[_] = -1; c[0] = i = 0; a[0] = c[1] = j = 1; a[1] = u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[i, k] == 0, BitAnd[j, k] == 0], k++]; If[k == u, While[c[u] > -1, u++]]; i = j; j = k; Set[{a[n], c[k]}, {Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ IntegerDigits[k, 2]], n}], {n, 2, nn}]; Array[a, nn + 1, 0]
import Data.List (delete)
a084937 n = a084937_list !! (n-1)
a084937_list = 1 : 2 : f 2 1 [3..] where
f x y zs = g zs where
g (u:us) | gcd y u > 1 || gcd x u > 1 = g us
| otherwise = u : f u x (delete u zs)
-- Reinhard Zumkeller, Jan 28 2012
N:= 1000: # to get a(n) until the first entry > N
a[1]:= 1: a[2]:= 2:
R:= {$3..N}:
for n from 3 while R <> {} do
success:= false;
for r in R do
if igcd(r,a[n-1]) = 1 and igcd(r,a[n-2])=1 then
a[n]:= r;
R:= R minus {r};
success:= true;
break
fi
od:
if not success then break fi;
od:
seq(a[i], i = 1 .. n-1); # Robert Israel, Dec 12 2014
lst={1,2,3}; unused=Range[4,100]; While[n=Select[unused, CoprimeQ[#, lst[[-1]]] && CoprimeQ[#, lst[[-2]]] &, 1]; n != {}, AppendTo[lst, n[[1]]]; unused=DeleteCases[unused, n[[1]]]]; lst
f[s_] := Block[{k = 1, l = Take[s, -2]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s, k], k++]; Append[s, k]]; Nest[f, {1, 2}, 67] (* Robert G. Wilson v, Jun 26 2011 *)
taken(k,t=v[k])=for(i=3,k-1, if(v[i]==t, return(1))); 0 step(k,g)=while(gcd(k,g)>1, k++); k first(n)=local(v=vector(n,i,i)); my(nxt=3,t); for(k=3,n, v[k]=step(nxt, t=v[k-1]*v[k-2]); while(taken(k), v[k]=step(v[k]+1,t)); if(v[k]==t, while(taken(k+1,t++),))); v \\ Charles R Greathouse IV, Aug 26 2016
from math import gcd A084937_list, l1, l2, s, b = [1,2], 2, 1, 3, set() for _ in range(10**3): i = s while True: if not i in b and gcd(i,l1) == 1 and gcd(i,l2) == 1: A084937_list.append(i) l2, l1 = l1, i b.add(i) while s in b: b.remove(s) s += 1 break i += 1 # Chai Wah Wu, Dec 09 2014
Table from _Walter Trump_, May 11 2022, showing initial terms of A353725 (column 1) and A353726 (column 3). The central column shows the corresponding entry of A353715 written in base 2. 0 1 0 1 110 2 2 1100 3 3 1101000 11 4 11110000 54 5 1111100000 74 6 11101000000 88 7 110110000000 183 12 11111000000000000 3913 13 10011111110000000000000 124845 16 111111111110000000000000000 2469947 17 1111111101100000000000000000 4005550 18 1011111111111000000000000000000 19917707
A353715(3) = 12 = 2^2*3, so a(3) = 2. A353715(11) = 104 = 2^3*13, so a(11) = 3.
from itertools import count, islice def A353724_gen(): # generator of terms s, a, b, c, ab = {0,1}, 0, 1, 2, 1 yield 0 while True: for n in count(c): if not (n & ab or n in s): yield len(t := bin(b+n))-len(t.rstrip('0')) a, b = b, n ab = a|b s.add(n) while c in s: c += 1 break A353724_list = list(islice(A353724_gen(),30)) # Chai Wah Wu, May 11 2022
The first 60 terms of A353710 are 0, / 1, / 2, / 3, 3, 3, / 5, 5, 5, 5, 5, 5, 5, 5, / 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, / 11, 11, 11, / 15, 15, 15, ... The slashes indicate the initial runs of lengths 1, 1, 1, 3, 8, 40, 3, ...
from itertools import count, islice def A353718_gen(): # generator of terms s, a, b, c, ab, k = {0,1}, 0, 1, 2, 1, 1 yield from (1,1) while True: for n in count(c): if not (n & ab or n in s): a, b = b, n ab = a|b s.add(n) if c in s: yield k k = 0 while c in s: c += 1 k += 1 break A353718_list = list(islice(A353718_gen(),20)) # Chai Wah Wu, May 10 2022
Table showing initial values of n (column 1) and a(n) (column 3). The central column shows the corresponding entry of A353715 written in base 2. The entries in column 2 end in exactly n zeros. n A353715(a(n)) a(n) 0 1 0 1 110 2 2 1100 3 3 1101000 11 4 11110000 54 5 1111100000 74 6 11101000000 88 7 110110000000 183 8 1111110101100000000 20334 9 10111110101000000000 30938 10 1111111110000000000 21247 11 1001111111100000000000 90575 12 11111000000000000 3913 13 10011111110000000000000 124845 14 111110111110100000000000000 2643790 15 10011111111111000000000000000 5828721 16 111111111110000000000000000 2469947 17 1111111101100000000000000000 4005550 18 1011111111111000000000000000000 19917707
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