A364297 a(n) = A348717(A163511(n)).

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, 20, 10, 14, 2, 64, 32, 162, 16, 108, 54, 250, 8, 72, 36, 150, 18, 100, 50, 98, 4, 48, 24, 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

Offset: 0

Author

Antti Karttunen, Aug 15 2023

Keywords

• nonn

Comments

For all i, j: a(i) = a(j) => A278531(i) = A278531(j).

As the underlying sequence A163511 can be represented as a binary tree, so can this be also:

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 20 10 14 2

etc.

Each rightward leaning branch stays constant, because a(2n+1) = a(n).

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

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Index entries for sequences related to binary expansion of n
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000265, A000668, A007283, A163511, A243071, A348717, A364949, A364956, A365423.

Cf. also A245611, A245612, A278531, A286531, A364959, A364963.

Programs

PARI

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));
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));
A364297(n) = A348717(A163511(n));

Formula

a(0) = 1, a(1) = 2, a(2n) = A163511(2n) = 2*A163511(n), and for n > 0, a(2n+1) = a(n).