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A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers. 6
1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 35, 37, 45, 49, 53, 61, 64, 69, 75, 81, 89, 91, 95, 113, 128, 131, 135, 141, 143, 145, 151, 161, 165, 169, 175, 207, 223, 225, 243, 245, 247, 251, 256, 265, 281, 299, 309, 311, 329, 343, 355, 359, 361, 375, 377, 385 (list; graph; refs; listen; history; text; internal format)
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 not necessarily strict antichains of multisets spanning an initial interval of positive integers.

LINKS

Table of n, a(n) for n=1..59.

EXAMPLE

The sequence of multisystems whose MM-numbers belong to the sequence begins:

   1: {}

   2: {{}}

   3: {{1}}

   4: {{},{}}

   7: {{1,1}}

   8: {{},{},{}}

   9: {{1},{1}}

  13: {{1,2}}

  15: {{1},{2}}

  16: {{},{},{},{}}

  19: {{1,1,1}}

  27: {{1},{1},{1}}

  32: {{},{},{},{},{}}

  35: {{2},{1,1}}

  37: {{1,1,2}}

  45: {{1},{1},{2}}

  49: {{1,1},{1,1}}

  53: {{1,1,1,1}}

  61: {{1,2,2}}

  64: {{},{},{},{},{},{}}

  69: {{1},{2,2}}

  75: {{1},{2},{2}}

  81: {{1},{1},{1},{1}}

  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[normQ[primeMS/@primeMS[#]], stableQ[primeMS[#], Divisible]]&]

CROSSREFS

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A318401, A319721, A320275, A320456.

Sequence in context: A032876 A032843 A023774 * A321699 A050271 A322742

Adjacent sequences:  A319834 A319835 A319836 * A319838 A319839 A319840

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 16 2018

STATUS

approved

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Last modified May 9 09:12 EDT 2021. Contains 343699 sequences. (Running on oeis4.)