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%I #13 Sep 23 2018 21:32:14
%S 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,
%T 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,
%U 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
%N Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.
%C 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.
%H Antti Karttunen, <a href="/A319004/b319004.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%F A001055(n) <= a(n) <= A074206(n). - _Antti Karttunen_, Sep 23 2018
%e The a(60) = 11 ordered factorizations:
%e (2*2*3*5),
%e (2*2*15), (2*3*10), (2*6*5), (4*3*5),
%e (2*30), (3*20), (4*15), (12*5), (6*10),
%e (60).
%e The a(60) = 11 ordered multiset partitions:
%e {{1,1,2,3}}
%e {{1},{1,2,3}}
%e {{2},{1,1,3}}
%e {{1,1,2},{3}}
%e {{1,1},{2,3}}
%e {{1,2},{1,3}}
%e {{1},{1},{2,3}}
%e {{1},{2},{1,3}}
%e {{1},{1,2},{3}}
%e {{1,1},{2},{3}}
%e {{1},{1},{2},{3}}
%t facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t lix[n_]:=LCM@@PrimePi/@If[n==1,{},FactorInteger[n]][[All,1]];
%t Table[Length[Select[Join@@Permutations/@facs[n],OrderedQ[lix/@#]&]],{n,100}]
%o (PARI)
%o is_weakly_increasing(v) = { for(i=2,#v,if(v[i]<v[i-1], return(0))); (1); };
%o A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
%o 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));
%o A319004(n) = if((1==n)||isprime(n),1,A319004aux(n, List([]))); \\ _Antti Karttunen_, Sep 23 2018
%Y Cf. A001055, A056239, A063834, A074206, A290103, A296150, A316223, A317545, A317546, A318559, A319002, A319003.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 07 2018
%E More terms from _Antti Karttunen_, Sep 23 2018
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