A263648 a(1)=4, a(2)=9: a(n) is the smallest semiprime not yet appearing in the sequence which is coprime to a(n-1) and not coprime to a(n-2).

4, 9, 10, 21, 22, 15, 14, 25, 6, 35, 26, 49, 34, 77, 38, 33, 46, 39, 58, 51, 62, 57, 74, 69, 82, 87, 86, 93, 94, 111, 106, 123, 118, 129, 122, 141, 134, 159, 142, 177, 146, 183, 158, 201, 166, 213, 178, 219, 194, 237, 202, 249, 206, 267, 214, 291, 218, 303, 226, 309

Offset: 1

Author

Bob Selcoe, Oct 22 2015

Keywords

• nonn

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.

Crossrefs

Cf. A098550, A119718, A000040 (primes), A100484 (even semiprimes), A001748 (essentially, semiprimes with smallest factor 3).

Programs

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  *)

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.

Extensions

More terms from Jean-François Alcover, Oct 06 2018