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A353730 a(1)=2; thereafter a(n) is the smallest positive number not yet used which is compatible with the condition that a(n) is relatively prime to the next n terms. 8
2, 1, 3, 4, 5, 7, 9, 11, 8, 13, 17, 19, 23, 25, 21, 29, 31, 37, 16, 41, 43, 47, 53, 59, 61, 67, 71, 73, 55, 79, 27, 49, 83, 89, 97, 101, 103, 107, 26, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 85, 121, 223 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Similar to A247665, which is obtained if the condition "smallest positive number" is changed to "smallest number >= 2".
It would be nice to have a proof that the numbers 6, 10, 12, 14, 15, 18, 20, 22, ... are missing from this sequence. It appears that the missing numbers are 6, 10, 12, 14, 15, 18, 20, 22, 24, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, ..., but since there is no proof that any one of these is really missing, this sequence cannot yet be added to the OEIS.
LINKS
Russ Cox, Table of n, a(n) for n = 1..100000 (terms 1..1000 from Alois P. Heinz; terms 1..10000 from Chai Wah Wu)
EXAMPLE
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.
PROG
(Python)
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
CROSSREFS
Cf. A247665; A353734 (powers of 2).
For the even terms, see A354146.
Sequence in context: A282864 A277518 A277383 * A077159 A277677 A075379
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 16 2022
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)