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

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, 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, 5, 3, 6, 1, 16, 1, 3, 4, 5, 2, 5, 1, 17, 8, 3, 1, 9, 3

Offset: 1

Author

Gus Wiseman, Aug 16 2022

Keywords

• nonn

Comments

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.

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}.

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.

Examples

			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.
		

Crossrefs

Numbers grouped by number of gaps in prime indices are A073491-A073495.

These are the standard composition numbers of rows of A356226.

Using Heinz numbers instead of standard compositions gives A356231.

Positions of first appearances are A356603, sorted A356232.

A001221 counts distinct prime factors, with sum A001414.

A003963 multiplies together the prime indices.

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

A066099 lists compositions in standard order.

A132747 counts non-isolated divisors, complement A132881.

A333627 represents the run-lengths of standard compositions.

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

Cf. A000120, A001222, A060680-A060683, A066205, A286470, A287170, A328166, A356227, A356228, A356229.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

Formula

A000120(a(n)) = A287170(n).

A333766(a(n)) = A356228(n).

A333768(a(n)) = A356227(n).