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.
from itertools import count, islice from math import gcd from collections import deque def A249559_gen(): # generator of terms aset, aqueue, c, f = {3}, deque([3]), 2, True yield 3 while True: for m in count(c): if m not in aset and all(gcd(m,a) == 1 for a in aqueue): yield m aset.add(m) aqueue.append(m) if f: aqueue.popleft() f = not f while c in aset: c += 1 break A249559_list = list(islice(A249559_gen(),50)) # Chai Wah Wu, May 19 2022
# from Nadia Heninger, Oct 28 2014: s is the starting point (2 in A247665, 3 here). def gen(s): sequence = [s] available = range(2, 2*s) available.pop(available.index(s)) yield s while True: available.extend(range(available[-1]+1, next_prime(available[-1])+1)) for i, e in enumerate(available): if all([gcd(e, sequence[j])==1 for j in range(-len(sequence)/2, 0)]): available.pop(i) sequence.append(e) yield(e) break g = gen(3) [g.next() for i in range(40)] # gets first 40 terms
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
nn = 120; c[] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, AllTrue[Array[a[i - #] &, Floor[i/2]], CoprimeQ[#, k] &]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 21 2022 *)
A090252_first(N, U=[0], L=List())=vector(N, i, for(k=U[1]+1, oo, setsearch(U, k) && next; foreach(L,m, gcd(k,m)>1 && next(2)); bitand(i,1) || listpop(L,1); listput(L,k); if( k>U[1]+1, U=setunion(U,[k]), U[1]++; while(#U>1 && U[2]==U[1]+1, U=U[^1]));break); L[#L]) \\ M. F. Hasler, Jun 14 2022
from math import gcd, prod
from itertools import count, islice
def agen(): # generator of terms
alst = [1]; aset = {1}; yield 1
mink = 2
for n in count(2):
k, prodall = mink, prod(alst[n-n//2-1:n-1])
while k in aset or gcd(prodall, k) != 1: k += 1
alst.append(k); aset.add(k); yield k
while mink in aset: mink += 1
print(list(islice(agen(), 64))) # Michael S. Branicky, May 21 2022
// See Links section.
# A354790 = Squarefree version of the Two-Up sequence A090252 # This produces 2*M terms in the array a # Assumes b117 is a list of sufficiently many squarefree numbers A005117 # Test if u is relatively prime to all of a[i], i = i1..i2 perpq:=proc(u,i1,i2) local i; global a; for i from i1 to i2 do if igcd(u,a[i])>1 then return(-1); fi; od: 1; end; a:=Array(1..10000,-1); hit:=Array(1..10000,-1); # 1 if i has appeared a[1]:=1; a[2]:=2; hit[1]:=1; hit[2]:=1; M:=100; M1 := 1000; for p from 2 to M do # step 1 want a[2p-1] relatively prime to a[p] ... a[2p-2] sw1:=-1; for j from 1 to M1 do c:=b117[j]; if hit[c] = -1 and perpq(c,p,2*p-2) = 1 then a[2*p-1]:=c; hit[c]:=1; sw1:=1; break; fi; od: # od j if sw1 = -1 then error("no luck, step 1, p =",p ); fi; # step 2 want a[2p] relatively prime to a[p] ... a[2p-1] sw2:=-1; for j from 1 to M1 do c:=b117[j]; if hit[c] = -1 and perpq(c,p,2*p-1) = 1 then a[2*p]:=c; hit[c]:=1; sw2:=1; break; fi; od: # od j if sw2 = -1 then error("no luck, step 2, p =",p ); fi; od: # od p [seq(a[i],i=1..2*M)];
nn = 60; pp[] = 1; k = r = 1; c[] = False; a[1] = 1; Do[Set[m, SelectFirst[Union@ Append[Times @@ # & /@ Subsets[#, {2, Infinity}], Prime[r]] &[Prime@ Select[Range[If[k == 1, r, k + 1]], p[Prime[#]] < n &]], ! c[#] &]]; Set[a[n], m]; (c[m] = True; If[PrimeQ[m], r++]; If[n > 1, Map[(Set[p[#], 2 n]; pp[#]++) &, #]]) &@ FactorInteger[m][[All, 1]]; While[pp[Prime[k]] > 2, k++], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Sep 06 2022 *)
\\ See Links section.
from math import lcm, gcd from itertools import count, islice from collections import deque from sympy import factorint def A354790_gen(): # generator of terms aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True yield 1 while True: for m in count(c): if m not in aset and gcd(m,b) == 1 and all(map(lambda n:n<=1,factorint(m).values())): yield m aset.add(m) aqueue.append(m) if f: aqueue.popleft() b = lcm(*aqueue) f = not f while c in aset: c += 1 break A354790_list = list(islice(A354790_gen(),30)) # Chai Wah Wu, Jul 17 2022
a(1) = 2 must be rel. prime to a(2), so a(2)=1. a(2) = 1 must be rel. prime to a(3) and a(4), so we can take them to be 3 and 4. a(3) = 3 must be rel. prime to a(5), a(6), so we can take them to be 5 and 7. a(4) = 4 must be rel. prime to a(7), a(8), so we can take them to be 9 and 11. At each step after the first, we must choose two new numbers, and we must make sure that not only are they rel. prime to a(n), they are also rel. prime to all a(i), i>n, that have been already chosen.
from itertools import count, islice from math import gcd from collections import deque def A353730_gen(): # generator of terms aset, aqueue, c, f = {2}, deque([2]), 1, True yield 2 while True: for m in count(c): if m not in aset and all(gcd(m,a) == 1 for a in aqueue): yield m aset.add(m) aqueue.append(m) if f: aqueue.popleft() f = not f while c in aset: c += 1 break A353730_list = list(islice(A353730_gen(),30)) # Chai Wah Wu, May 18-19 2022
from math import gcd, prod
from itertools import count, islice
def agen(): # generator of terms
alst, aset, mink = [1], {1}, 2
for n in count(2):
k, s = mink, n - n//2
prodall = prod(alst[n-n//2-1:n-1])
while k in aset or gcd(prodall, k) != 1: k += 1
alst.append(k); aset.add(k)
if k%2 == 0: yield k
while mink in aset: mink += 1
print(list(islice(agen(), 9))) # Michael S. Branicky, May 23 2022
Comments