OFFSET
1,1
COMMENTS
Contrary to what might be expected (see comment after Proof), a(n) is not a permutation of all semiprimes; it is a permutation of even semiprimes {S_2} and semiprimes with smallest factor 3 {S_3}, plus {25, 35, 49, 77}. Proof (Start):
i. The sequence is infinite: we can always consider p*q for a(n+1), where p is the smallest factor in a(n-1) and q is the smallest prime > than the largest factor of any term already appearing in the sequence;
ii. a(15)=38 = 2*19 = 2*prime(8) and a(16)=33 = 3*11 = 3*prime(5);
iii. all {S_2} <= 38 and {S_3} <= 33 have appeared up to a(16), with 38 and 33 being maximum terms in {S_2} and {S_3}, respectively;
iv. all semiprimes with smallest factor >= 5 which are < 2*prime(9)=46 and 3*prime(6)=39 have appeared up to a(16). Consequently, the terms starting at a(17)=46 alternate between 2*prime(k) and 3*prime(k-3) k=9..infinity.
v. the only other numbers to have appeared are {25, 35, 49, 77}.
(End)
The above behavior is in contrast to A119718 (a permutation of all semiprimes because it lacks the constraint of a(n) being not coprime to a(n-2)). Interestingly, this sequence (A263648) shares the same essential rules as A098550 (the Yellowstone permutation) and many of its variations, while A119718 does not; one therefore might expect the opposite behavior to occur between this sequence and A119718. What observations or generalizations might we draw from this?
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015. Also Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.7
Michael De Vlieger, Annotated scatterplot of (n, k), where k is the position of a(n) in A001358, for n = 1..48. Color code: even a(n) in red, 3 | a(n) in blue, 6 | a(n) in purple, else black.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^18, showing even terms in red, 3 | a(n) in blue, 6 | a(n) in purple, else black. The first 15 terms are labeled.
FORMULA
For n >= 15:
a(n) = 2*prime((n+1)/2) when n is odd;
a(n) = 3*prime(n/2-3) when n is even.
MATHEMATICA
a[1]=4; a[2]=9; a[n_] := a[n] = Module[{k}, For[k=6, True, k++, If[MatchQ[ FactorInteger[k], {{_, 1}, {_, 1}}|{{_, 2}}] && FreeQ[Array[a, n-1], k] && CoprimeQ[k, a[n-1]] && !CoprimeQ[k, a[n-2]], Return[k]]]]; Array[a, 60] (* Jean-François Alcover, Oct 06 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Bob Selcoe, Oct 22 2015
EXTENSIONS
More terms from Jean-François Alcover, Oct 06 2018
STATUS
approved