A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

2, 3, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 151, 223, 247, 251, 281, 299, 311, 359, 377, 427, 463, 503, 593, 611, 659, 689, 703, 719, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1291, 1321, 1339, 1363, 1511, 1619, 1703, 1733, 1739, 1757, 1769

Offset

1,1

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 connected strict antichains of multisets spanning an initial interval of positive integers.

Links

Table of n, a(n) for n=1..51.

Example

The sequence of multisystems whose MM-numbers belong to the sequence begins:
    2: {{}}
    3: {{1}}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}

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]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
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], Length[zsm[primeMS[#]]]==1]&]

Crossrefs

Cf. A003963, A006126, A055932, A056239, A112798, A285573, A286520, A293994, A302242, A318401, A319719, A319837, A320275, A320456, A320532.

Sequence in context: A331948 A342529 A172238 * A038940 A019383 A249749

Adjacent sequences: A319493 A319494 A319495 * A319497 A319498 A319499

Keyword

nonn

Author

Gus Wiseman, Dec 16 2018

Status

approved