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

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, 73, 79, 83, 85, 89, 14, 97, 101, 103, 33, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 65, 233

Offset: 1

Author

Michael De Vlieger and N. J. A. Sloane, Jul 17 2022

Keywords

• nonn

Comments

A version of the Two-Up sequence A090252 that is restricted to squarefree numbers.

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Thomas Scheuerle, Comments on A354790 regarding the possibility of composites with more than two factors.
• Rémy Sigrist, Table of n, a(n) for n = 1..100000
• Rémy Sigrist, PARI program with comments
• Rémy Sigrist, C program (inspired by Russ Cox's Go program for A247665)
• Michael De Vlieger, Compact annotated plot of prime p | A354790(n) at (n, pi(p)) for composite A354790(n), 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.
• Michael De Vlieger, Comprehensive annotated plot of prime p | A354790(n) at (n, pi(p)) for composite A354790(n), n <= 10^5. Color function indicates the number k > 1 of appearances of divisor p in the sequence.

Crossrefs

Cf. A090252, A247665, A354169.

See A354791 and A354792 for the nonprime terms.

See A355895 for the even terms.

Programs

C

// See Links section.

Maple

# 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)];

Mathematica

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 *)

PARI

\\ See Links section.

Python

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

Extensions

More terms from Rémy Sigrist, Jul 17 2022