A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

1, 2, 3, 7, 13, 15, 19, 35, 37, 53, 61, 69, 89, 91, 95, 113, 131, 141, 143, 145, 151, 161, 165, 223, 247, 251, 265, 281, 299, 309, 311, 329, 355, 359, 377, 385, 407, 427, 437, 463, 503, 591, 593, 611, 655, 659, 667, 671, 689, 703, 719, 721, 759, 791, 827, 851

Offset

1,2

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of strict antichains of multisets spanning an initial interval of positive integers.

Links

Table of n, a(n) for n=1..56.

Example

The sequence of multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   7: {{1,1}}
  13: {{1,2}}
  15: {{1},{2}}
  19: {{1,1,1}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  53: {{1,1,1,1}}
  61: {{1,2,2}}
  69: {{1},{2,2}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[200], And[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], stableQ[primeMS[#], Divisible]]&]

Crossrefs

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A319496, A319721, A319837, A320275, A320456.

Sequence in context: A273056 A026472 A286176 * A322703 A225098 A101739

Adjacent sequences: A318398 A318399 A318400 * A318402 A318403 A318404

Keyword

nonn

Author

Gus Wiseman, Dec 16 2018

Status

approved