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A061775
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Number of nodes in rooted tree with Matula-Goebel number n.
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143
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1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 6, 6, 7, 6, 7, 6, 6, 7, 6, 6, 7, 6, 7, 7, 6, 6, 7, 7, 6, 7, 6, 7, 8, 7, 7, 7, 7, 8, 7, 7, 6, 8, 8, 7, 7, 7, 6, 8, 7, 7, 8, 7, 8, 8, 6, 7, 8, 8, 7, 8, 7, 7, 9, 7, 8, 8, 7, 8, 9, 7, 7, 8, 8, 7, 8, 8, 7, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 7, 8, 8, 8, 9, 7, 7, 9
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OFFSET
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1,2
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COMMENTS
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Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration).
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LINKS
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FORMULA
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a(1) = 1; if n = p_t (= the t-th prime), then a(n) = 1+a(t); if n = uv (u,v>=2), then a(n) = a(u)+a(v)-1.
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EXAMPLE
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a(4) = 3 because the rooted tree corresponding to the Matula-Goebel number 4 is "V", which has one root-node and two leaf-nodes, three in total.
See also the illustrations in A061773.
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MAPLE
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with(numtheory): a := proc (n) local u, v: u := n-> op(1, factorset(n)): v := n-> n/u(n): if n = 1 then 1 elif isprime(n) then 1+a(pi(n)) else a(u(n))+a(v(n))-1 end if end proc: seq(a(n), n = 1..108); # Emeric Deutsch, Sep 19 2011
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MATHEMATICA
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a[n_] := Module[{u, v}, u = FactorInteger[#][[1, 1]]&; v = #/u[#]&; If[n == 1, 1, If[PrimeQ[n], 1+a[PrimePi[n]], a[u[n]]+a[v[n]]-1]]]; Table[a[n], {n, 108}] (* Jean-François Alcover, Jan 16 2014, after Emeric Deutsch *)
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PROG
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(Haskell)
import Data.List (genericIndex)
a061775 n = genericIndex a061775_list (n - 1)
a061775_list = 1 : g 2 where
g x = y : g (x + 1) where
y = if t > 0 then a061775 t + 1 else a061775 u + a061775 v - 1
where t = a049084 x; u = a020639 x; v = x `div` u
(PARI)
A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs, t, i); pfs=factor(n); pfs[, 1]=apply(t->A061775(t), pfs[, 1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i, 1]*pfs[i, 2])}));
for(n=1, 10000, write("b061775.txt", n, " ", A061775(n)));
(Python)
from functools import lru_cache
from sympy import isprime, factorint, primepi
@lru_cache(maxsize=None)
if n == 1: return 1
if isprime(n): return 1+A061775(primepi(n))
return 1+sum(e*(A061775(p)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022
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CROSSREFS
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Sum of entries in row n of irregular table A214573.
Cf. A005517 (the position of the first occurrence of n).
Cf. A005518 (the position of the last occurrence of n).
Cf. A091233 (their difference plus one).
Cf. A214572 (Numbers k such that a(k) = 8).
Cf. also A000081, A061773, A049084, A020639, A049076, A078442, A091238, A091204, A091205, A109082, A127301, A109129, A193402, A193405, A193406, A196047, A196068, A198333, A206487, A206494, A206496, A214569, A214571, A213670, A214568, A228599, A245817, A245818.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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