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A320275 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers. 6
2, 3, 7, 9, 13, 19, 27, 37, 49, 53, 61, 81, 89, 91, 113, 131, 151, 169, 223, 243, 247, 251, 281, 299, 311, 343, 359, 361, 377, 427, 463, 503, 593, 611, 637, 659, 689, 703, 719, 729, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1183, 1291, 1321, 1339 (list; graph; refs; listen; history; text; internal format)
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 not necessarily strict connected antichains of multisets spanning an initial interval of positive integers.

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

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

    2: {{}}

    3: {{1}}

    7: {{1,1}}

    9: {{1},{1}}

   13: {{1,2}}

   19: {{1,1,1}}

   27: {{1},{1},{1}}

   37: {{1,1,2}}

   49: {{1,1},{1,1}}

   53: {{1,1,1,1}}

   61: {{1,2,2}}

   81: {{1},{1},{1},{1}}

   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}}

  169: {{1,2},{1,2}}

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[normQ[primeMS/@primeMS[#]], stableQ[primeMS[#], Divisible], Length[zsm[primeMS[#]]]==1]&]

CROSSREFS

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A286520, A290103, A293994, A302242, A316476, A319496, A319837, A320456, A320532.

Sequence in context: A262775 A092293 A215485 * A096072 A305121 A014837

Adjacent sequences:  A320272 A320273 A320274 * A320276 A320277 A320278

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 16 2018

STATUS

approved

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Last modified May 27 20:36 EDT 2020. Contains 334667 sequences. (Running on oeis4.)