A336737 Number of factorizations of n whose factors have pairwise intersecting prime signatures.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 7, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 6, 3, 2, 1, 9, 2, 2, 2

Offset

1,4

Comments

First differs from A327400 at a(72) = 9, A327400(72) = 10.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Links

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

Example

The a(n) factorizations for n = 2, 4, 12, 24, 30, 36, 60:
  (2)  (4)    (12)     (24)       (30)     (36)       (60)
       (2*2)  (2*6)    (2*12)     (5*6)    (4*9)      (2*30)
              (2*2*3)  (2*2*6)    (2*15)   (6*6)      (3*20)
                       (2*2*2*3)  (3*10)   (2*18)     (5*12)
                                  (2*3*5)  (3*12)     (6*10)
                                           (2*3*6)    (2*5*6)
                                           (2*2*3*3)  (2*2*15)
                                                      (2*3*10)
                                                      (2*2*3*5)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
prisig[n_]:=If[n==1, {}, Last/@FactorInteger[n]];
Table[Length[Select[facs[n], stableQ[#, Intersection[prisig[#1], prisig[#2]]=={}&]&]], {n, 100}]

Crossrefs

A001055 counts factorizations.

A118914 is sorted prime signature.

A124010 is prime signature.

A336736 counts factorizations with disjoint signatures.

Cf. A003182, A051185, A305843, A305844, A305854, A306006, A319752, A319787, A319789, A321469, A336424.

Sequence in context: A038548 A320732 A305149 * A327400 A323303 A068108

Adjacent sequences: A336734 A336735 A336736 * A336738 A336739 A336740

Keyword

nonn

Author

Gus Wiseman, Aug 06 2020

Status

approved