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A352187 a(1)=1, a(2)=2; thereafter, a(n) is the smallest number m not yet in the sequence such that gcd(m,a(n-1)) > 1 and gcd(m,a(n-2))=1, except that the second condition is ignored if it would imply that no choice for m were possible. 13
1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 25, 30, 22, 11, 33, 27, 36, 26, 13, 39, 42, 28, 32, 34, 17, 51, 45, 35, 49, 56, 38, 19, 57, 48, 40, 55, 77, 63, 54, 44, 121, 66, 46, 23, 69, 60, 50, 52, 91, 105, 72, 58, 29, 87, 75, 65, 104, 62, 31, 93, 78, 64, 68, 85, 95, 76, 74, 37, 111, 81, 84, 70, 115, 207, 96, 80 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is an intermediate sequence between the EKG sequence A064413 (which only involves the first condition) and the Enots Wolley sequence A336957 (which involves both conditions).

By ignoring the second condition when necessary we guarantee that a(n) always exists (because it always exists for the EKG sequence), and so no backtracking is needed in this sequence.

The second condition is ignored precisely when every prime that divides a(n-1) also divides a(n-2).

Conjecture 1: Every positive number appears.

Conjecture 2: The primes are the slowest numbers to appear. That is, when a prime p appears here, all numbers less than p have already appeared. To put this another way, the record high points in A352188 ("when does n appear") are exactly the primes (and 1).

Conjecture 3: When a prime p first divides some term, that term is 2*p, which is followed by p then 3p. [It seems possible that we could see 2*a, 6*b, 3*p, p, 2*p. This does not happen in the first 10000 terms, but I can't prove it never happens. I can prove that every prime divides some term, and that the primes appear in increasing order. Conjecture 3 might be refuted by a more extensive calculation.]

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

MAPLE

# To produce the first 1000 terms:

with(numtheory):

omega := proc(n) nops(numtheory[factorset](n)) end proc:

hit:=Array(1..100000, 0);

M:=100000;

a:=[1, 2]; K:=1; L:=2; hit[1]:=1; hit[2]:=2;

for n from 3 to 1000 do

sw1:=0;

# find a[n]

if factorset(L) subset factorset(K) then

# use EKG rule

for i from 1 to M do

if hit[i]=0 and igcd(i, L)>1 then a:=[op(a), i]; K:=L; L:=i; hit[i]:=n; sw1:=1; break; fi;

od:

if sw1=0 then error("failed EKG, n, i =", n, i); fi;

else

# use Enots Wolley rule

for i from 1 to M do

if hit[i]=0 and igcd(i, L)>1 and igcd(i, K)=1 then a:=[op(a), i]; K:=L; L:=i; hit[i]:=n; sw1:=1; break; fi;

od:

if sw1=0 then error("failed WOLLEY, n, i =", n, i); fi;

fi:

od:

a;

PROG

(Python)

from math import gcd

from itertools import count, islice

from sympy import primefactors

def A352187_gen(): # generator of terms

bset, blist, mmax = {1, 2}, [1, 2], 3

yield from blist

while True:

for m in count(mmax):

if gcd(m, blist[-1]) > 1 and m not in bset:

if all(blist[-2] % p == 0 for p in primefactors(blist[-1])) or gcd(m, blist[-2]) == 1:

yield m

blist = [blist[-1], m]

bset.add(m)

while mmax in bset:

mmax += 1

break

A352187_list = list(islice(A352187_gen(), 20)) # Chai Wah Wu, Mar 14 2022

CROSSREFS

Cf. A064413, A336957, A352188 (when n appears), A352189, A352190.

Records: A352191, A352192.

Sequence in context: A350927 A096665 A064413 * A357963 A357994 A357942

Adjacent sequences: A352184 A352185 A352186 * A352188 A352189 A352190

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mar 12 2022

STATUS

approved

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Last modified February 6 16:43 EST 2023. Contains 360110 sequences. (Running on oeis4.)