A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 3, 2, 1, 1, 3, 1, 5, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 5, 2, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 6

Offset: 1

Author

Gus Wiseman, Aug 10 2022

Keywords

• nonn
• tabf

Comments

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}.

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

			Triangle  begins: {}, {1}, {1}, {2}, {1}, {2}, {1}, {3}, {2}, {1,1}, {1}, {3}, {1}, {1,1}, {2}, {4}, {1}, {3}, {1}, {2,1}, ... For example, the prime indices of 20 are {1,1,3}, which separates into maximal gapless submultisets {{1,1},{3}}, so row 20 is (2,1).
The prime indices of 18564 are {1,1,2,4,6,7}, which separates into {1,1,2}, {4}, {6,7}, so row 18564 is (3,1,2). This corresponds to the factorization 18564 = 12 * 7 * 221.
		

Crossrefs

Row sums are A001222.

Singleton row positions are A073491, complement A073492.

Length-2,3,4 row positions are A073493-A073495.

Row lengths are A287170, firsts A066205.

Row minima are A356227.

Row maxima are A356228.

Bisected run-lengths are A356229.

Standard composition numbers of rows are A356230.

Heinz numbers of rows are A356231.

Positions of first appearances are A356232.

A001221 counts distinct prime factors, with sum A001414.

A001223 lists the prime gaps, reduced A028334.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

A132747 counts non-isolated divisors, complement A132881.

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

Cf. A000005, A055874, A060680-A060683, A137921, A193829, A286470, A328166.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]