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A235120 a(n) = b((n+1)/2)^2 if n is odd and a(n) = b(n/2)*b(n/2+1) if n is even, where b = A057452.

%I A235120 #20 Nov 24 2019 23:49:55
%S A235120 64,152,361,1273,4489,22177,109561,735151,4932841,43480517,383258929,
%T A235120 4299363701,48229869769,668098103693,9254743549921,154323135646769,
%U A235120 2573343071840641,50602620151819037,995057827403377609,22686465423182125223,517231963031027272681
%N A235120 a(n) = b((n+1)/2)^2 if n is odd and a(n) = b(n/2)*b(n/2+1) if n is even, where b = A057452.
%C A235120 a(n) is the Matula number of the rooted tree obtained by identifying the roots of the trees Q(floor((n+1)/2)) and Q(ceiling((n+1)/2)) defined in A057452.
%D A235120 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A235120 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011
%H A235120 E. Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H A235120 F. Goebel, <a href="http://dx.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 A235120 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A235120 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.
%p A235120 b := proc (n) option remember: if n = 1 then 8 else ithprime(b(n-1)) end if: end proc: a := proc (n) if `mod`(n, 2) = 1 then b((1/2)*n+1/2)^2 else b((1/2)*n)*b((1/2)*n+1) end if end proc: seq(a(n), n = 1 .. 17);
%t A235120 b[1] = 8; b[n_] := b[n] = Prime[b[n-1]];
%t A235120 a[n_] := If[OddQ[n], b[(n+1)/2]^2, b[n/2]*b[n/2 + 1]];
%t A235120 Array[a, 21] (* _Jean-François Alcover_, Nov 26 2017, from Maple *)
%Y A235120 Cf. A057452.
%K A235120 nonn
%O A235120 1,1
%A A235120 _Emeric Deutsch_, Jan 18 2014