%I #37 Jun 25 2024 17:06:47
%S 1,3,6,6,10,10,11,11,15,15,15,17,17,17,21,20,17,25,20,24,24,21,25,30,
%T 28,25,36,28,24,34,21,37,28,24,32,44,30,30,34,41,25,40,28,32,48,36,34,
%U 55,37,45,32,40,37,64,36,49,41,34,24,59,44,28,57,70,44,44,30,37,48,53,41,81,40,44,63,49,41,56,32,74
%N Number of subtrees in the rooted tree with Matula-Goebel number n.
%C 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.
%H É. Czabarka, L. Székely, and S. Wagner, <a href="https://doi.org/10.1016/j.dam.2009.07.004">The inverse problem for certain tree parameters</a>, Discrete Appl. Math., 157, 2009, 3314-3319.
%H Emeric Deutsch, <a href="https://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H F. Goebel, <a href="https://doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H I. Gutman and A. Ivic, <a href="https://doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H D. W. Matula, <a href="https://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F Let b(n)=A184160(n) denote the number of those subtrees of the rooted tree with Matula-Goebel number n that contain the root. Then a(1)=1; if n=prime(t) (=the t=th prime), then a(n)=1+a(t)+b(t); if n=r*s (r,s,>=2), then a(n)=a(r)+a(s)+(b(r)-1)*(b(s)-1)-1. The Maple program is based on this recursive formula.
%e a(4) = 6 because the rooted tree with Matula-Goebel number 4 is V; it has 6 subtrees (three 1-vertex subtrees, two 1-edge subtrees, and the tree itself).
%p with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+b(pi(n)) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+1 else a(r(n))+a(s(n))+(b(r(n))-1)*(b(s(n))-1)-1 end if end proc: seq(a(n), n = 1 .. 80);
%t r[n_] := FactorInteger[n][[1, 1]];
%t s[n_] := n/r[n];
%t b[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + b[PrimePi[n]], True, b[r[n]]*b[s[n]]];
%t a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]] + b[PrimePi[n]] + 1, True, a[r[n]] + a[s[n]] + (b[r[n]] - 1)*(b[s[n]] - 1) - 1];
%t Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Jun 24 2024, after Maple code *)
%Y Cf. A184160 (subtrees containing the root), A184164 (numbers not occurring as terms).
%K nonn
%O 1,2
%A _Emeric Deutsch_, Oct 19 2011