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

1,3

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.

Links

Table of n, a(n) for n=1..87.

Example

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.

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}]

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.

Sequence in context: A104145 A230981 A123675 * A322872 A356227 A356228

Adjacent sequences: A356223 A356224 A356225 * A356227 A356228 A356229

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 10 2022

Status

approved