A354225 Lexicographically earliest sequence of distinct positive integers such that a(1) = 1 and for any n > 1, n / gcd(n, a(n)) and a(n) / gcd(n, a(n)) are prime.
1, 3, 2, 6, 7, 4, 5, 12, 15, 14, 13, 8, 11, 10, 9, 24, 19, 27, 17, 28, 33, 26, 29, 16, 35, 22, 18, 20, 23, 42, 37, 48, 21, 38, 25, 54, 31, 34, 51, 56, 43, 30, 41, 52, 63, 58, 53, 32, 77, 70, 39, 44, 47, 36, 65, 40, 69, 46, 61, 84, 59, 74, 45, 96, 55, 78, 71
Offset: 1
Keywords
Examples
The first terms are: n a(n) g=gcd(n, a(n)) n/g a(n)/g -- ---- -------------- --- ------ 1 1 1 1 1 2 3 1 2 3 3 2 1 3 2 4 6 2 2 3 5 7 1 5 7 6 4 2 3 2 7 5 1 7 5 8 12 4 2 3 9 15 3 3 5 10 14 2 5 7 11 13 1 11 13 12 8 4 3 2 13 11 1 13 11 14 10 2 7 5
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Annotated log-log plot of a(n), n = 1..2^14, showing records in red, local minima in blue, highlighting primes in green, fixed points in gold, and composite prime powers in magenta.
- Index entries for sequences that are permutations of the natural numbers
Crossrefs
Cf. A122280.
Programs
-
Mathematica
nn = 120; c[] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, AllTrue[{i/#, k/#}, PrimeQ] &@ GCD[i, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 22 2022 *)
-
PARI
s=0; for (n=1, 67, for (v=1, oo, if (!bittest(s, v) && (n==1 || (isprime(n/g=gcd(n,v)) && isprime(v/g))), print1 (v", "); s+=2^v; break)))
Formula
a(prime(2*n)) = prime(2*n-1) (where prime(n) denotes the n-th prime number).
Comments