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This sequence The shortest of these factorizations is A356233listed at A356234, length A287170.

The shortest (maximal) of these factorizations is listed at A356234, length A287170.

A000005 counts divisors.

A001055 counts factorizations.

A001221 counts distinct prime factors, sum A001414.

A003963 multiplies together the prime indices.

A132747 counts non-isolated divisors, complement A132881.

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

A001055 counts factorizations.

A001221 counts distinct prime factors, sum A001414.

A003963 multiplies together the prime indices.

A132747 counts non-isolated divisors, complement A132881.

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

`Cf. A000005, A001222, A060680-A060683, `A066205, A073491-A073495, A193829, A330103, A356234A328195, A328335-A356237A328458.

Cf. A328195 maxlen_divchn_consec_divs_grtr1, A328335 consec_prix_relpri, A328336 no_consec_relpri, A328457 maxrun_divs_grtr1, A328458 max_divrun_nontriv.

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. We define a number to be gapless (listed by A066311) iff its prime indices cover an interval of positive integers.

Gapless numbers (listed by A066311) are characterized by having prime indices covering an interval of positive integers.

allocated for Gus WisemanNumber of integer factorizations of n into gapless numbers (A066311).

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 2, 5, 1, 4, 1, 2, 1, 1, 1, 7, 2, 1, 3, 2, 1, 4, 1, 7, 1, 1, 2, 9, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 12, 2, 2, 1, 2, 1, 7, 1, 3, 1, 1, 1, 8, 1, 1, 2, 11, 1, 2, 1, 2, 1, 2, 1, 16, 1, 1, 4, 2, 2, 2, 1, 5, 5, 1, 1, 4, 1, 1

1,4

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.

Gapless numbers (listed by A066311) are characterized by having prime indices covering an interval of positive integers.

The counted factorizations of n = 2, 4, 8, 12, 24, 36, 48:

(2) (4) (8) (12) (24) (36) (48)

(2*2) (2*4) (2*6) (3*8) (4*9) (6*8)

(2*2*2) (3*4) (4*6) (6*6) (2*24)

(2*2*3) (2*12) (2*18) (3*16)

(2*2*6) (3*12) (4*12)

(2*3*4) (2*2*9) (2*3*8)

(2*2*2*3) (2*3*6) (2*4*6)

(3*3*4) (3*4*4)

(2*2*3*3) (2*2*12)

(2*2*2*6)

(2*2*3*4)

(2*2*2*2*3)

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

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

sqq[n_]:=Max@@Differences[primeMS[n]]<=1;

Table[Length[Select[facs[n], And@@sqq/@#&]], {n, 100}]

This sequence is A356233.

The shortest (maximal) of these factorizations is listed at A356234, length A287170.

A356226 lists the lengths of maximal gapless submultisets of prime indices:

- length: A287170

- minimum: A356227

- maximum: A356228

- bisected length: A356229

- standard composition: A356230

- Heinz number: A356231

- positions of first appearances: A356232

A001055 counts factorizations.

A001221 counts distinct prime factors, sum A001414.

A003963 multiplies together the prime indices.

A132747 counts non-isolated divisors, complement A132881.

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

`Cf. A000005, A001222, A060680-A060683, `A066205, A073491-A073495, A193829, A330103, A356234-A356237.

Cf. A328195 maxlen_divchn_consec_divs_grtr1, A328335 consec_prix_relpri, A328336 no_consec_relpri, A328457 maxrun_divs_grtr1, A328458 max_divrun_nontriv.

allocated

nonn

Gus Wiseman, Aug 28 2022

approved

editing

allocated for Gus Wiseman

allocated

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