OFFSET
1,2
COMMENTS
Let i,j,k,h be any 4 consecutive terms, then there must be a prime p; p|j*k but p!|i, for if not then for any prime q; q|j*k implies q|i. But then for any h such that gcd(h,j*k) > 1 we would have gcd(h,i) > 1; contradiction since h is a term. Any number of the form m = p*x where p|j*k, p!|i and x is any novel prime not dividing i must be a candidate for a(n) since then gcd(m,j*k) > 1 and gcd(m,i) = 1. Therefore the sequence is well defined and infinite. Similar to the Yellowstone sequence A098550, from which it diverges at a(8) = 5.
Conjecture: Sequence is a permutation of the positive integers, with primes in order.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, with a color function showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple additionally represents powerful numbers that are not prime powers.
Michael De Vlieger, Plot p^m | n at (x,y) = (n,pi(p)), n = 1..2048, showing multiplicity m = 1 in black, m = 2 in red, m = 3 in orange, ..., m = 10 in magenta.
EXAMPLE
Given a(1,2,3) = 1,2,3 a(4) = 4 since 4 it is the smallest novel number which shares a prime factor with 6 but not with 1. This forces a(5) = 9, the smallest number prime to 2 but not to 12.
a(21,22,23) = 11,55,20 so a(24) must be a multiple of either 2 or 5 since both divide 55*20 but not 11. Using 5 we would get a(24) = 30 or if using 2 we would have a(24) = 24. Thus a(24) = 24.
MATHEMATICA
nn = 120; c[_] := False;
Set[{h, i, j, u}, Range[4]];
{h, i, j}~Join~Reap[Do[
k = u; While[Or[c[k], CoprimeQ[i*j, k], ! CoprimeQ[h, k]], k++];
Set[{c[k], h, i, j}, {True, i, j, k}]; Sow[k];
If[k == u, While[c[u], u++]],
{n, 4, nn}] ][[-1, 1]] (* Michael De Vlieger, Dec 19 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Dec 19 2024
STATUS
approved