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%I #6 Aug 20 2022 18:43:55
%S 1,1,1,1,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,2,2,2,1,2,2,1,2,2,1,2,1,2,2,
%T 2,1,2,2,2,2,2,2,2,2,1,2,2,1,2,2,2,2,2,1,3,2,2,2,2,1,2,2,2,1,3,2,2,2,
%U 2,2,2,1,2,2,1,2,2,2,2,2,1,2,2,2,3,2,2
%N Number of maximal gapless submultisets of the prime indices of 2n.
%C A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.
%C 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.
%C This is a bisection of A287170, but is important in its own right because the even numbers are exactly those whose prime indices begin with 1.
%F a(n) = A287170(2n).
%e The prime indices of 2*9282 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(9282) = 3.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Length[Split[primeMS[2n],#1>=#2-1&]],{n,100}]
%Y This is the even (bisected) case of A287170, firsts A066205.
%Y Alternate row-lengths of A356226, minima A356227(2n), maxima A356228(2n).
%Y A001221 counts distinct prime factors, sum A001414.
%Y A001222 counts prime indices, listed by A112798, sum A056239.
%Y A003963 multiplies together the prime indices of n.
%Y A073093 counts the prime indices of 2n.
%Y A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
%Y Cf. A000005, A060680, A060681, A132747, A132881, A286470, A289509, A356230, A356231, A356232.
%K nonn
%O 1,5
%A _Gus Wiseman_, Aug 16 2022
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