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%I #34 Sep 06 2023 06:58:32
%S 1,2,4,2,8,4,6,2,16,8,18,4,12,6,10,2,32,16,54,8,36,18,50,4,24,12,30,6,
%T 20,10,14,2,64,32,162,16,108,54,250,8,72,36,150,18,100,50,98,4,48,24,
%U 90,12,60,30,70,6,40,20,42,10,28,14,22,2,128,64,486,32,324,162,1250,16,216,108,750,54,500,250,686,8,144
%N a(n) = A348717(A163511(n)).
%C For all i, j: a(i) = a(j) => A278531(i) = A278531(j).
%C As the underlying sequence A163511 can be represented as a binary tree, so can this be also:
%C 1
%C |
%C ...................2...................
%C 4 2
%C 8......../ \........4 6......../ \........2
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C 16 8 18 4 12 6 10 2
%C 32 16 54 8 36 18 50 4 24 12 30 6 20 10 14 2
%C etc.
%C Each rightward leaning branch stays constant, because a(2n+1) = a(n).
%C Conjecture: Mersenne primes (A000668) gives all such odd numbers k for which a(k) = A348717(k). If true, then it immediately implies that map n -> A163511(n) [or equally: map n -> A243071(n)] has no other fixed points than those given by A007283. But see also A364959. - Edited Sep 03 2023
%H Antti Karttunen, <a href="/A364297/b364297.txt">Table of n, a(n) for n = 0..8192</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(0) = 1, a(1) = 2, a(2n) = A163511(2n) = 2*A163511(n), and for n > 0, a(2n+1) = a(n).
%o (PARI)
%o A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
%o A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
%o A364297(n) = A348717(A163511(n));
%Y Cf. A000265, A000668, A007283, A163511, A243071, A348717, A364949, A364956, A365423.
%Y Cf. also A245611, A245612, A278531, A286531, A364959, A364963.
%K nonn
%O 0,2
%A _Antti Karttunen_, Aug 15 2023
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