%I #7 Aug 21 2022 14:13:31

%S 1,2,2,3,2,3,2,5,3,4,2,5,2,4,3,7,2,5,2,6,4,4,2,7,3,4,5,6,2,5,2,11,4,4,

%T 3,7,2,4,4,10,2,6,2,6,5,4,2,11,3,6,4,6,2,7,4,10,4,4,2,7,2,4,6,13,4,6,

%U 2,6,4,6,2,11,2,4,5,6,3,6,2,14,7,4,2,10

%N Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%C 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 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 A001222(a(n)) = A287170(n).

%F A055396(a(n)) = A356227(n).

%F A061395(a(n)) = A356228(n).

%e 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), with Heinz number 30, so a(18564) = 30.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Times@@Prime/@Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]

%Y Positions of prime terms are A073491, complement A073492.

%Y Positions of terms with bigomega 2-4 are A073493-A073495.

%Y Applying bigomega gives A287170, firsts A066205, even bisection A356229.

%Y These are the Heinz numbers of the rows of A356226.

%Y Minimal/maximal prime indices are A356227/A356228.

%Y A version for standard compositions is A356230, firsts A356232/A356603.

%Y A001221 counts distinct prime factors, with sum A001414.

%Y A003963 multiplies together the prime indices.

%Y A056239 adds up the prime indices, row sums of A112798.

%Y A132747 counts non-isolated divisors, complement A132881.

%Y A356069 counts gapless divisors, initial A356224 (complement A356225).

%Y Cf. A000005, A001222, A055932, A060680-A060683, A193829, A286470, A328166, A356233-A356237.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 18 2022