cp's OEIS Frontend

A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

%I A356231 #7 Aug 21 2022 14:13:31
%S A356231 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 A356231 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 A356231 2,6,4,6,2,11,2,4,5,6,3,6,2,14,7,4,2,10
%N A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.
%C A356231 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 A356231 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 A356231 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 A356231 A001222(a(n)) = A287170(n).
%F A356231 A055396(a(n)) = A356227(n).
%F A356231 A061395(a(n)) = A356228(n).
%e A356231 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 A356231 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356231 Table[Times@@Prime/@Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]
%Y A356231 Positions of prime terms are A073491, complement A073492.
%Y A356231 Positions of terms with bigomega 2-4 are A073493-A073495.
%Y A356231 Applying bigomega gives A287170, firsts A066205, even bisection A356229.
%Y A356231 These are the Heinz numbers of the rows of A356226.
%Y A356231 Minimal/maximal prime indices are A356227/A356228.
%Y A356231 A version for standard compositions is A356230, firsts A356232/A356603.
%Y A356231 A001221 counts distinct prime factors, with sum A001414.
%Y A356231 A003963 multiplies together the prime indices.
%Y A356231 A056239 adds up the prime indices, row sums of A112798.
%Y A356231 A132747 counts non-isolated divisors, complement A132881.
%Y A356231 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356231 Cf. A000005, A001222, A055932, A060680-A060683, A193829, A286470, A328166, A356233-A356237.
%K A356231 nonn
%O A356231 1,2
%A A356231 _Gus Wiseman_, Aug 18 2022