A252941 - OEIS

%I #36 Mar 30 2015 21:52:34

%S 1,2,3,2,3,2,3,5,2,7,5,2,3,5,2,3,5,7,2,7,2,5,3,7,2,5,3,2,11,3,13,11,

%T 13,3,11,2,13,3,5,2,7,3,13,2,17,2,3,5,17,2,3,5,11,2,17,5,13,2,3,7,13,

%U 2,3,5,7,2,19,3,7,19,2,3,7,5,19,2,11

%N Irregular triangle T(n,k) read by rows: T(1,1) = 1, otherwise row n lists the prime factors of A098550(n), with duplicates omitted.

%C Row n contains the distinct prime factors of A098550(n), in increasing order. For example, when n=13, A098550(13) = 35 and T(13,k) = [5,7].

%C Because A098550 is a permutation of the natural numbers, this sequence is infinite and contains every prime infinitely often.

%C Primes appear in order; that is, first appearance of prime(j) occurs prior to first appearance of prime(j+1).

%C T(n,1) = A251101(n), which are the smallest prime factors of A098550(n), n>1.

%C For n>1, let each coefficient in T(n,1) be prime(i). The ratio that each coefficient appears in T(j,1) {j=1..n} approaches A038110(i)/A038111(i) as j increases.  For example, prime(4) = 7: as j increases, the ratio that 7 appears in T(j,1) approaches 4/105, because A038110(4)/A038111(4) = 4/105.

%H David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, <a href="http://arxiv.org/abs/1501.01669">The Yellowstone Permutation</a>, arXiv preprint arXiv:1501.01669, 2015.

%e Triangle begins T(1,1):

%e 1

%e 2

%e 3

%e 2

%e 3

%e 2

%e 3 5

%e 2 7

%e 5

%e 2 3

%e 5

%e 2 3

%e 5 7

%e 2

%e 7

%e 2 5

%e 3 7

%e 2 5

%e 3

%e 2 11

%e e.g., n=13: A098550(13) = 35; T(13,k) = 5,7.

%Y Cf. A098550.

%K nonn,tabf

%O 1,2

%A _Bob Selcoe_, Mar 22 2015