cp's OEIS Frontend

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

%I A356227 #8 Aug 13 2022 22:24:52
%S A356227 0,1,1,2,1,2,1,3,2,1,1,3,1,1,2,4,1,3,1,1,1,1,1,4,2,1,3,1,1,3,1,5,1,1,
%T A356227 2,4,1,1,1,1,1,1,1,1,3,1,1,5,2,1,1,1,1,4,1,1,1,1,1,4,1,1,1,6,1,1,1,1,
%U A356227 1,1,1,5,1,1,3,1,2,1,1,1,4,1,1,1,1,1,1
%N A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.
%C A356227 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 A356227 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 A356227 a(n) = A333768(A356230(n)).
%F A356227 a(n) = A055396(A356231(n)).
%e A356227 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) = 1.
%t A356227 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356227 Table[If[n==1,0,Min@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]
%Y A356227 Positions of first appearances are A000079.
%Y A356227 The maximal gapless submultisets are counted by A287170, firsts A066205.
%Y A356227 These are the row-minima of A356226, firsts A356232.
%Y A356227 The greatest instead of smallest size is A356228.
%Y A356227 A001221 counts distinct prime factors, with sum A001414.
%Y A356227 A001222 counts prime factors with multiplicity.
%Y A356227 A001223 lists the prime gaps, reduced A028334.
%Y A356227 A003963 multiplies together the prime indices of n.
%Y A356227 A056239 adds up prime indices, row sums of A112798.
%Y A356227 A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
%Y A356227 A356224 counts even gapless divisors, complement A356225.
%Y A356227 Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A356229.
%K A356227 nonn
%O A356227 1,4
%A A356227 _Gus Wiseman_, Aug 13 2022