This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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.
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
-- Reinhard Zumkeller, Sep 03 2013
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
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 *)
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))); \\ Antti Karttunen, Aug 16 2014
from functools import lru_cache from sympy import isprime, factorint, primepi @lru_cache(maxsize=None) def A061775(n): 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
The first 15 rows of this irregular triangular table:
0,
1,
2, 2,
3, 3, 3,
3, 4, 4, 3,
4, 4, 5, 4, 4,
4, 5, 5, 5, 5, 4,
4, 5, 6, 5, 6, 5, 4,
4, 5, 6, 6, 6, 6, 5, 4,
5, 5, 6, 6, 7, 6, 6, 5, 5,
5, 6, 6, 6, 7, 7, 6, 6, 6, 5,
4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4,
5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5,
5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5,
6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6
etc.
E.g., we have
a(1) = T(0,0) = a(0) + a(0) + 1 = 1,
a(2) = T(1,0) = a(1) + a(0) + 1 = 2,
a(3) = T(0,1) = a(0) + a(1) + 1 = 2,
a(4) = T(2,0) = a(2) + a(0) + 1 = 3, etc.
up_to = 105; A002260(n) = (n-binomial((sqrtint(8*n)+1)\2, 2)); \\ From A002260 A004736(n) = (1-n+(n=sqrtint(8*n)\/2)*(n+1)\2); \\ From A004736 A071673list(up_to) = { my(v=vector(1+up_to)); v[1] = 0; for(n=1,up_to,v[1+n] = 1 + v[A004736(n)] + v[A002260(n)]); (v); }; v071673 = A071673list(up_to); A071673(n) = v071673[1+n]; \\ Antti Karttunen, Aug 17 2021
(define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))
The rooted plane trees encoded here are: .....................o...............o.........o...o..o....... .....................|...............|..........\./...|....... .......o....o...o....o....o.o.o..o...o.o.o.o.o...o....o...o... .......|.....\./.....|.....\|/....\./...\|.|/....|.....\./.... *......*......*......*......*......*......*......*......*..... 1......2......3......4......5......6......7......8......9.....
a(5) = 7, as 5 is the 3rd prime and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(32) = a(2^5) = A048723(A014580(1),a(5)) = A048723(2,7) = 128. a(48) = a(3 * 2^4) = 3 X A048723(2,a(4+1)-1) = 3 X A048723(2,7-1) = 3 X 64 = 192.
a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3) and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^(a(4+1)-1) = 3 * 2^(9-1) = 3 * 256 = 768.
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