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A336107
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Number of permutations of the prime indices of n with at least one non-singleton run, or non-separations.
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12
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0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 6, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 9, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 6, 0, 0, 0
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OFFSET
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1,12
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COMMENTS
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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.
A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
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LINKS
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FORMULA
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EXAMPLE
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The a(n) non-separations for n = 12, 36, 60, 72, 180, 420:
(11) (112) (1122) (1123) (11122) (11223) (11234)
(211) (1221) (1132) (11212) (11232) (11243)
(2112) (2113) (11221) (11322) (11324)
(2211) (2311) (12112) (12213) (11342)
(3112) (12211) (12231) (11423)
(3211) (21112) (13122) (11432)
(21121) (13221) (21134)
(21211) (21123) (21143)
(22111) (21132) (23114)
(22113) (23411)
(22131) (24113)
(22311) (24311)
(23112) (31124)
(23211) (31142)
(31122) (32114)
(31221) (32411)
(32112) (34112)
(32211) (34211)
(41123)
(41132)
(42113)
(42311)
(43112)
(43211)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Permutations[primeMS[n]], MatchQ[#, {___, x_, x_, ___}]&]], {n, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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