OFFSET
1,2
COMMENTS
The sequence contains groups of consecutive primes separated by groups of composite terms. See the attached images of the first 100000 and 25 million terms.
The fixed points are 1, 2, 3, 4, 15, 51, 63, 363, 437, ... There are thirty-two fixed points in the first 100000 terms and it is likely there are infinitely many in total. The sequence is conjectured to be a permutation of the positive integers.
Theorem: This sequence is a permutation of the positive integers, and the primes appear in their natural order. See link for proof. - N. J. A. Sloane, Sep 24 2024 and Oct 02 2024
LINKS
Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, with a color function showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not prime powers.
Scott R. Shannon, Image of the first 100000 terms. The terms are colored red, yellow, green, blue, violet if they have one, two, three, four, or five or more prime factors. The thin white line is a(n) = n.
Scott R. Shannon, Image of the first 25 million terms [Conventions as in preceding image.]
N. J. A. Sloane, Proof that this sequence is a permutation of the positive integers
N. J. A. Sloane, Table of n, a(n) for n = 1..100000
EXAMPLE
a(8) = 7 as a(7) = 5 is a prime number, and 7 is the smallest unused number that is coprime to 5.
Comment from N. J. A. Sloane, Oct 02 2024 (Start)
The following table was calculated by Scott R. Shannon on Sep 29 2024. It shows the beginning, end, and length of the k-th run of successive primes.
2 8 5 : 9 7 2
3 18 11 : 21 19 4
4 40 23 : 45 43 6
5 84 47 : 92 83 9
6 162 89 : 177 167 16
7 321 173 : 351 349 31
8 649 353 : 702 683 54
9 1286 691 : 1379 1361 94
10 2550 1367 : 2724 2699 175
11 5096 2707 : 5412 5387 317
12 10188 5393 : 10787 10739 600
13 20406 10753 : 21502 21467 1097
14 40883 21481 : 42958 42863 2076
15 81932 42899 : 85791 85717 3860
16 164190 85733 : 171441 171103 7252
17 328490 171131 : 342216 341729 13727
18 657509 341743 : 683462 682811 25954
19 1316258 682819 : 1365580 1364747 49323
20 2635513 1364761 : 2729447 2728129 93935
21 5276876 2728163 : 5456194 5454167 179319
22 10565366 5454181 : 10908253 10906271 342888
23 21155215 10906297 : 21812343 21808453 657129
24 42355195 21808483 : 43616683 43611721 1261489
25 84797387 43611731 : 87223016 87215467 2425630
26 169759097 87215483 : 174430392 174419101 4671296
(End)
MATHEMATICA
nn = 120; c[_] := False; Do[Set[{a[i], c[i]}, {i, True}], {i, 2}];
j = a[2]; u = 3;
Do[k = u; If[PrimeQ[j],
While[Or[c[k], GCD[j, k] > 1], k++],
While[Or[c[k], CoprimeQ[j, k]], k++]];
Set[{a[i], c[k], j}, {k, True, k}];
If[k == u, While[c[u], u++]], {i, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 29 2024 *)
PROG
(Python)
from itertools import islice
from math import gcd
from sympy import isprime
def A375564_gen(): # generator of terms
aset, a, b = {1, 2}, 2, 3
yield from (1, 2)
while True:
c = b
if isprime(a):
while c in aset or gcd(c, a)>1:
c+=1
else:
while c in aset or gcd(c, a)==1:
c+=1
aset.add(c)
yield (a:=c)
while b in aset:
b += 1
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Scott R. Shannon, Aug 19 2024
STATUS
approved