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A184155
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The Matula-Goebel number of rooted trees having all leaves at the same level.
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9
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1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 49, 53, 57, 59, 63, 64, 67, 73, 81, 83, 85, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 159, 171, 189, 227, 241, 243, 256, 269, 277, 289, 307, 311, 331, 335, 343, 361, 365, 367, 371, 391, 393, 399, 419, 425, 431, 439, 441, 477
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OFFSET
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1,2
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COMMENTS
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The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
The sequence is infinite.
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REFERENCES
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F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
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LINKS
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FORMULA
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In A184154 one constructs for each n the generating polynomial P(n,x) of the leaves of the rooted tree with Matula-Goebel number n, according to their levels. The Maple program finds those n (between 1 and 500) for which P(n,x) is a monomial.
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EXAMPLE
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7 is in the sequence because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having all leaves at level 2.
2^m is in the sequence for each positive integer m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
Sequence of trees begins:
01 o
02 (o)
03 ((o))
04 (oo)
05 (((o)))
07 ((oo))
08 (ooo)
09 ((o)(o))
11 ((((o))))
16 (oooo)
17 (((oo)))
19 ((ooo))
21 ((o)(oo))
23 (((o)(o)))
25 (((o))((o)))
27 ((o)(o)(o))
31 (((((o)))))
(End)
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MAPLE
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with(numtheory): P := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: A := {}: for n to 500 do if degree(numer(subs(x = 1/x, P(n)))) = 0 then A := `union`(A, {n}) else end if end do: A;
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MATHEMATICA
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primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
dep[n_]:=If[n===1, 0, 1+Max@@dep/@primeMS[n]];
rnkQ[n_]:=And[SameQ@@dep/@primeMS[n], And@@rnkQ/@primeMS[n]];
Select[Range[2000], rnkQ] (* Gus Wiseman, Mar 30 2018 *)
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CROSSREFS
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Cf. A000081, A003238, A004111, A007097, A048816, A061775, A109082, A184154, A214577, A244925, A276625, A290689, A290760, A290822, A298422, A298424, A298426.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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