A319004 Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 8, 1, 5, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 16, 2, 2, 2, 11, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 16, 2, 5, 2, 4, 1, 12, 2, 8, 2, 2, 1, 11, 1, 2, 4, 32, 2, 5, 1, 4, 2, 5, 1, 23, 1, 2, 4, 4, 2, 5, 1, 16, 8, 2, 1, 11, 2, 2, 2, 8, 1, 12, 2, 4, 2, 2, 2, 32, 1, 5, 4, 11, 1, 5, 1, 8, 5

Offset

1,4

Comments

Also the number of ordered multiset partitions of the multiset of prime indices of n where the sequence of LCMs of the parts is weakly increasing. If we form a multiorder by treating integer partitions (a,...,z) as multiarrows LCM(a,...,z) <= {z,...,a}, then a(n) is the number of triangles whose composite ground is the integer partition with Heinz number n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to Heinz numbers

Formula

A001055(n) <= a(n) <= A074206(n). - Antti Karttunen, Sep 23 2018

Example

The a(60) = 11 ordered factorizations:
  (2*2*3*5),
  (2*2*15), (2*3*10), (2*6*5), (4*3*5),
  (2*30), (3*20), (4*15), (12*5), (6*10),
  (60).
The a(60) = 11 ordered multiset partitions:
     {{1,1,2,3}}
    {{1},{1,2,3}}
    {{2},{1,1,3}}
    {{1,1,2},{3}}
    {{1,1},{2,3}}
    {{1,2},{1,3}}
   {{1},{1},{2,3}}
   {{1},{2},{1,3}}
   {{1},{1,2},{3}}
   {{1,1},{2},{3}}
  {{1},{1},{2},{3}}

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
lix[n_]:=LCM@@PrimePi/@If[n==1, {}, FactorInteger[n]][[All, 1]];
Table[Length[Select[Join@@Permutations/@facs[n], OrderedQ[lix/@#]&]], {n, 100}]

Prog

(PARI)
is_weakly_increasing(v) = { for(i=2, #v, if(v[i]<v[i-1], return(0))); (1); };
A290103(n) = lcm(apply(p->primepi(p), factor(n)[, 1]));
A319004aux(n, facs) = if(1==n, is_weakly_increasing(apply(f -> A290103(f), Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1), newfacs = List(facs); listput(newfacs, d); s += A319004aux(n/d, newfacs))); (s));
A319004(n) = if((1==n)||isprime(n), 1, A319004aux(n, List([]))); \\ Antti Karttunen, Sep 23 2018

Crossrefs

Cf. A001055, A056239, A063834, A074206, A290103, A296150, A316223, A317545, A317546, A318559, A319002, A319003.

Sequence in context: A328892 A296131 A345344 * A234541 A066389 A077191

Adjacent sequences: A319001 A319002 A319003 * A319005 A319006 A319007

Keyword

nonn

Author

Gus Wiseman, Sep 07 2018

Extensions

More terms from Antti Karttunen, Sep 23 2018

Status

approved