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A319004
Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.
4
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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 07 2018
EXTENSIONS
More terms from Antti Karttunen, Sep 23 2018
STATUS
approved