A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

36, 72, 90, 100, 108, 126, 144, 180, 196, 198, 200, 216, 225, 234, 252, 270, 288, 300, 306, 324, 342, 350, 360, 378, 392, 396, 400, 414, 432, 441, 450, 468, 484, 500, 504, 522, 525, 540, 550, 558, 576, 588, 594, 600, 612, 630, 648, 650, 666, 675, 676, 684, 700

Offset

1,1

Comments

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.

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).

Links

Table of n, a(n) for n=1..53.

Wikipedia, Permutation pattern

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Example

The sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   72: {1,1,1,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  126: {1,2,2,4}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  196: {1,1,4,4}
  198: {1,2,2,5}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  234: {1,2,2,6}
  252: {1,1,2,2,4}
  270: {1,2,2,2,3}
  288: {1,1,1,1,1,2,2}
  300: {1,1,2,3,3}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Select[Permutations[primeMS[#]], MatchQ[#, {___, x_, ___, y_, ___, x_, ___}/; x<y]&&MatchQ[#, {___, x_, ___, y_, ___, x_, ___}/; x>y]&]!={}&]

Crossrefs

Replacing "and" with "or" gives A126706.

Positions of nonzero terms in A335462.

Permutations of prime indices are counted by A008480.

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

STC-numbers of permutations of prime indices are A333221.

Patterns matched by standard compositions are counted by A335454.

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

Sequence in context: A260919 A247381 A249726 * A192026 A036785 A338539

Adjacent sequences: A335460 A335461 A335462 * A335464 A335465 A335466

Keyword

nonn

Author

Gus Wiseman, Jun 20 2020

Status

approved