A356233 Number 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

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..86.

Example

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)

Mathematica

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}]

Crossrefs

The shortest 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).

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

Cf. A001222, A060680-A060683, A073491-A073495, A193829, A328195, A328335-A328458.

Sequence in context: A356228 A284257 A318883 * A353382 A231719 A123400

Adjacent sequences: A356230 A356231 A356232 * A356234 A356235 A356236

Keyword

nonn

Author

Gus Wiseman, Aug 28 2022

Status

approved