cp's OEIS Frontend

A356230 The a(n)-th composition in standard order is the sequence of lengths of maximal gapless submultisets of the prime indices of n.

%I A356230 #7 Aug 20 2022 23:20:32
%S A356230 0,1,1,2,1,2,1,4,2,3,1,4,1,3,2,8,1,4,1,5,3,3,1,8,2,3,4,5,1,4,1,16,3,3,
%T A356230 2,8,1,3,3,9,1,5,1,5,4,3,1,16,2,6,3,5,1,8,3,9,3,3,1,8,1,3,5,32,3,5,1,
%U A356230 5,3,6,1,16,1,3,4,5,2,5,1,17,8,3,1,9,3
%N A356230 The a(n)-th composition in standard order is the sequence of lengths of maximal gapless submultisets of the prime indices of n.
%C A356230 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%C A356230 A multiset 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 A356230 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 A356230 A000120(a(n)) = A287170(n).
%F A356230 A333766(a(n)) = A356228(n).
%F A356230 A333768(a(n)) = A356227(n).
%e A356230 The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}. These have lengths (3,1,2), which is the 38th composition in standard order, so a(18564) = 38.
%t A356230 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356230 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A356230 Table[stcinv[Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]
%Y A356230 Numbers grouped by number of gaps in prime indices are A073491-A073495.
%Y A356230 These are the standard composition numbers of rows of A356226.
%Y A356230 Using Heinz numbers instead of standard compositions gives A356231.
%Y A356230 Positions of first appearances are A356603, sorted A356232.
%Y A356230 A001221 counts distinct prime factors, with sum A001414.
%Y A356230 A003963 multiplies together the prime indices.
%Y A356230 A056239 adds up the prime indices, row sums of A112798.
%Y A356230 A066099 lists compositions in standard order.
%Y A356230 A132747 counts non-isolated divisors, complement A132881.
%Y A356230 A333627 represents the run-lengths of standard compositions.
%Y A356230 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356230 Cf. A000120, A001222, A060680-A060683, A066205, A286470, A287170, A328166, A356227, A356228, A356229.
%K A356230 nonn
%O A356230 1,4
%A A356230 _Gus Wiseman_, Aug 16 2022