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%I #6 Aug 14 2022 10:20:28
%S 0,1,1,2,1,2,1,3,2,1,1,3,1,1,2,4,1,3,1,2,1,1,1,4,2,1,3,2,1,3,1,5,1,1,
%T 2,4,1,1,1,3,1,2,1,2,3,1,1,5,2,2,1,2,1,4,1,3,1,1,1,4,1,1,2,6,1,2,1,2,
%U 1,2,1,5,1,1,3,2,2,2,1,4,4,1,1,3,1,1,1
%N Greatest size of a gapless submultiset of the prime indices of n.
%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 intervals: {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.
%F a(n) = A333766(A356230(n)).
%F a(n) = A061395(A356231(n)).
%e The prime indices of 700 are {1,1,3,3,4}, with maximal gapless submultisets {1,1}, {3,3,4}, so a(700) = 3.
%e The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[If[n==1,0,Max@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]
%Y Positions of first appearances are A000079.
%Y The maximal gapless submultisets are counted by A287170, firsts A066205.
%Y These are the row-maxima of A356226, firsts A356232.
%Y The smallest instead of greatest size is A356227.
%Y A001221 counts distinct prime factors, with sum A001414.
%Y A001222 counts prime factors with multiplicity.
%Y A001223 lists the prime gaps, reduced A028334.
%Y A003963 multiplies together the prime indices of n.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
%Y A356069 counts gapless divisors.
%Y A356224 counts even gapless divisors, complement A356225.
%Y Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A328162, A328457, A356229.
%K nonn
%O 1,4
%A _Gus Wiseman_, Aug 13 2022
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