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A354790 a(n) is the least positive squarefree number not already used that is coprime to the previous floor(n/2) terms.

%I A354790 #89 Aug 03 2024 11:14:08
%S A354790 1,2,3,5,7,11,6,13,17,19,23,29,31,35,22,37,39,41,43,47,53,59,61,67,71,
%T A354790 73,79,83,85,89,14,97,101,103,33,107,109,113,127,131,137,139,149,151,
%U A354790 157,163,167,173,179,181,191,193,197,199,211,223,227,229,65,233
%N A354790 a(n) is the least positive squarefree number not already used that is coprime to the previous floor(n/2) terms.
%C A354790 A version of the Two-Up sequence A090252 that is restricted to squarefree numbers.
%H A354790 Rémy Sigrist, <a href="/A354790/b354790.txt">Table of n, a(n) for n = 1..10000</a>
%H A354790 Thomas Scheuerle, <a href="/A354790/a354790_7.txt">Comments on A354790</a> regarding the possibility of composites with more than two factors.
%H A354790 Rémy Sigrist, <a href="/A354790/a354790_1.txt">Table of n, a(n) for n = 1..100000</a>
%H A354790 Rémy Sigrist, <a href="/A354790/a354790_8.txt">PARI program with comments</a>
%H A354790 Rémy Sigrist, <a href="/A354790/a354790.txt">C program</a> (inspired by Russ Cox's Go program for A247665)
%H A354790 Michael De Vlieger, <a href="/A354790/a354790.png">Compact annotated plot of prime p | A354790(n) at (n, pi(p)) for composite A354790(n)</a>, n <= 1500. Color function indicates the number k > 1 of appearances of divisor p in the sequence. Diagram supports a proposition similar to Conjecture 3 in A090252 but regarding this sequence. Indices n connected in red appear in A355897.
%H A354790 Michael De Vlieger, <a href="/A354790/a354790_1.png">Comprehensive annotated plot of prime p | A354790(n) at (n, pi(p)) for composite A354790(n)</a>, n <= 10^5. Color function indicates the number k > 1 of appearances of divisor p in the sequence.
%p A354790 # A354790 = Squarefree version of the Two-Up sequence A090252
%p A354790 # This produces 2*M terms in the array a
%p A354790 # Assumes b117 is a list of sufficiently many squarefree numbers A005117
%p A354790 # Test if u is relatively prime to all of a[i], i = i1..i2
%p A354790 perpq:=proc(u,i1,i2) local i; global a;
%p A354790 for i from i1 to i2 do if igcd(u,a[i])>1 then return(-1); fi; od: 1; end;
%p A354790 a:=Array(1..10000,-1);
%p A354790 hit:=Array(1..10000,-1); # 1 if i has appeared
%p A354790 a[1]:=1; a[2]:=2; hit[1]:=1; hit[2]:=1;
%p A354790 M:=100; M1 := 1000;
%p A354790 for p from 2 to M do
%p A354790 # step 1 want a[2p-1] relatively prime to a[p] ... a[2p-2]
%p A354790 sw1:=-1;
%p A354790 for j from 1 to M1 do
%p A354790 c:=b117[j];
%p A354790 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;
%p A354790 od: # od j
%p A354790 if sw1 = -1 then error("no luck, step 1, p =",p ); fi;
%p A354790 # step 2 want a[2p] relatively prime to a[p] ... a[2p-1]
%p A354790 sw2:=-1;
%p A354790 for j from 1 to M1 do
%p A354790 c:=b117[j];
%p A354790 if hit[c] = -1 and perpq(c,p,2*p-1) = 1 then a[2*p]:=c; hit[c]:=1; sw2:=1; break; fi;
%p A354790 od: # od j
%p A354790 if sw2 = -1 then error("no luck, step 2, p =",p ); fi;
%p A354790 od: # od p
%p A354790 [seq(a[i],i=1..2*M)];
%t A354790 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 *)
%o A354790 (PARI) \\ See Links section.
%o A354790 (Python)
%o A354790 from math import lcm, gcd
%o A354790 from itertools import count, islice
%o A354790 from collections import deque
%o A354790 from sympy import factorint
%o A354790 def A354790_gen(): # generator of terms
%o A354790     aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
%o A354790     yield 1
%o A354790     while True:
%o A354790         for m in count(c):
%o A354790             if m not in aset and gcd(m,b) == 1 and all(map(lambda n:n<=1,factorint(m).values())):
%o A354790                 yield m
%o A354790                 aset.add(m)
%o A354790                 aqueue.append(m)
%o A354790                 if f: aqueue.popleft()
%o A354790                 b = lcm(*aqueue)
%o A354790                 f = not f
%o A354790                 while c in aset:
%o A354790                     c += 1
%o A354790                 break
%o A354790 A354790_list = list(islice(A354790_gen(),30)) # _Chai Wah Wu_, Jul 17 2022
%o A354790 (C) // See Links section.
%Y A354790 Cf. A090252, A247665, A354169.
%Y A354790 See A354791 and A354792 for the nonprime terms.
%Y A354790 See A355895 for the even terms.
%K A354790 nonn
%O A354790 1,2
%A A354790 _Michael De Vlieger_ and _N. J. A. Sloane_, Jul 17 2022
%E A354790 More terms from _Rémy Sigrist_, Jul 17 2022