%I #12 Feb 09 2021 01:55:52

%S 36,72,90,100,108,126,144,180,196,198,200,216,225,234,252,270,288,300,

%T 306,324,342,350,360,378,392,396,400,414,432,441,450,468,484,500,504,

%U 522,525,540,550,558,576,588,594,600,612,630,648,650,666,675,676,684,700

%N Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

%C A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.

%C We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

%e The sequence of terms together with their prime indices begins:

%e 36: {1,1,2,2}

%e 72: {1,1,1,2,2}

%e 90: {1,2,2,3}

%e 100: {1,1,3,3}

%e 108: {1,1,2,2,2}

%e 126: {1,2,2,4}

%e 144: {1,1,1,1,2,2}

%e 180: {1,1,2,2,3}

%e 196: {1,1,4,4}

%e 198: {1,2,2,5}

%e 200: {1,1,1,3,3}

%e 216: {1,1,1,2,2,2}

%e 225: {2,2,3,3}

%e 234: {1,2,2,6}

%e 252: {1,1,2,2,4}

%e 270: {1,2,2,2,3}

%e 288: {1,1,1,1,1,2,2}

%e 300: {1,1,2,3,3}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Select[Permutations[primeMS[#]],MatchQ[#,{___,x_,___,y_,___,x_,___}/;x<y]&&MatchQ[#,{___,x_,___,y_,___,x_,___}/;x>y]&]!={}&]

%Y Replacing "and" with "or" gives A126706.

%Y Positions of nonzero terms in A335462.

%Y Permutations of prime indices are counted by A008480.

%Y Unsorted prime signature is A124010. Sorted prime signature is A118914.

%Y STC-numbers of permutations of prime indices are A333221.

%Y Patterns matched by standard compositions are counted by A335454.

%Y Cf. A056239, A056986, A112798, A158005, A181796, A333175, A335451, A335452, A335460, A335465.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 20 2020