A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

2, 3, 4, 5, 6, 7, 8, 9, 2, 5, 11, 12, 13, 2, 7, 15, 16, 17, 18, 19, 4, 5, 3, 7, 2, 11, 23, 24, 25, 2, 13, 27, 4, 7, 29, 30, 31, 32, 3, 11, 2, 17, 35, 36, 37, 2, 19, 3, 13, 8, 5, 41, 6, 7, 43, 4, 11, 45, 2, 23, 47, 48, 49, 2, 25, 3, 17, 4, 13, 53, 54, 5, 11, 8

Offset

1,1

Comments

Row-products are the positive integers 1, 2, 3, ...

Links

Table of n, a(n) for n=1..74.

Example

The first 16 rows:
   1 =
   2 = 2
   3 = 3
   4 = 4
   5 = 5
   6 = 6
   7 = 7
   8 = 8
   9 = 9
  10 = 2 * 5
  11 = 11
  12 = 12
  13 = 13
  14 = 2 * 7
  15 = 15
  16 = 16
The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.

Mathematica

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

Crossrefs

Row-lengths are A287170, firsts A066205, even bisection A356229.

Applying bigomega to all parts gives A356226, statistics A356227-A356232.

A001055 counts factorizations.

A001221 counts distinct prime factors, sum A001414.

A003963 multiplies together the prime indices.

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

A132747 counts non-isolated divisors, complement A132881.

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

Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.

Sequence in context: A043269 A122481 A110805 * A284255 A284253 A318492

Adjacent sequences: A356231 A356232 A356233 * A356235 A356236 A356237

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 28 2022

Status

approved