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A290760
Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).
26
1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, 3390, 4290, 7878, 9570, 10230, 11310, 13026, 15510, 15990, 18330, 26070, 30966, 37290, 39390, 40890, 44070, 45210, 65130, 84810, 94830, 98310, 104610, 122070, 124410, 132990, 154830, 159330, 175890, 198330, 201630
OFFSET
1,2
COMMENTS
A rooted tree is transitive if every terminal subtree is a branch of the root. A finitary set is transitive if every element is also a subset.
EXAMPLE
Let o = {}. The sequence of transitive finitary sets begins:
1 o
2 {o}
6 {o,{o}}
30 {o,{o},{{o}}}
78 {o,{o},{o,{o}}}
330 {o,{o},{{o}},{{{o}}}}
390 {o,{o},{{o}},{o,{o}}}
870 {o,{o},{{o}},{o,{{o}}}}
1410 {o,{o},{{o}},{{o},{{o}}}}
3198 {o,{o},{o,{o}},{{o,{o}}}}
3390 {o,{o},{{o}},{o,{o},{{o}}}}
4290 {o,{o},{{o}},{{{o}}},{o,{o}}}
7878 {o,{o},{o,{o}},{o,{o,{o}}}}
9570 {o,{o},{{o}},{{{o}}},{o,{{o}}}}
10230 {o,{o},{{o}},{{{o}}},{{{{o}}}}}
11310 {o,{o},{{o}},{o,{o}},{o,{{o}}}}
13026 {o,{o},{o,{o}},{{o},{o,{o}}}}
15510 {o,{o},{{o}},{{{o}}},{{o},{{o}}}}
15990 {o,{o},{{o}},{o,{o}},{{o,{o}}}}
18330 {o,{o},{{o}},{o,{o}},{{o},{{o}}}}
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
finitaryQ[n_]:=finitaryQ[n]=Or[n===1, With[{m=primeMS[n]}, {UnsameQ@@m, finitaryQ/@m}]/.List->And];
subprimes[n_]:=If[n===1, {}, Union@@Cases[FactorInteger[n], {p_, _}:>FactorInteger[PrimePi[p]][[All, 1]]]];
transitaryQ[n_]:=Divisible[n, Times@@subprimes[n]];
nn=100000; Fold[Select, Range[nn], {finitaryQ, transitaryQ}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 19 2017
STATUS
approved