%I #26 Jul 10 2020 22:06:53

%S 0,1,3,2,15,6,7,4,5,30,63,12,255,14,29,8,511,10,1023,60,13,126,2047,

%T 24,23,510,9,28,4095,58,31,16,125,1022,27,20,16383,2046,509,120,32767,

%U 26,65535,252,57,4094,262143,48,11,46,1021,1020,1048575,18,119,56,2045,8190,2097151,116,4194303,62,25,32,503,250,8388607,2044,4093,54,16777215,40

%N Mersenne-prime fixing variant of A243071: a(n) = A243071(A332213(n)).

%C Any Mersenne prime (A000668) times any power of 2 (i.e., 2^k, for k>=0) is fixed by this sequence, including also all even perfect numbers.

%C From _Antti Karttunen_, Jul 10 2020: (Start)

%C This is a "tuned variant" of A243071, and has many of the same properties.

%C For example, for n > 1, A007814(a(n)) = A007814(n) - A209229(n), that is, this map preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is decremented by one, and in particular, a(2^k * n) = 2^k * a(n) for all n > 1. Also, like A243071, this bijection maps primes to the terms of A000225 (binary repunits). However, the "tuning" (A332213) has a specific effect that each Mersenne prime (A000668) is mapped to itself. Therefore the terms of A335431 are fixed by this map. Furthermore, I conjecture that there are no other fixed points. For the starters, see the proof in A335879, which shows that at least none of the terms of A335882 are fixed.

%C (End)

%H Antti Karttunen, <a href="/A332215/b332215.txt">Table of n, a(n) for n = 1..8192</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(n) = A243071(A332213(n)).

%F For all n >= 1, a(A335431(n)) = A335431(n), a(A335882(n)) = A335879(n). - _Antti Karttunen_, Jul 10 2020

%o (PARI) A332215(n) = A243071(A332213(n));

%Y Cf. A243071, A332210, A332213, A332214 (inverse permutation), A335431 (conjectured to be all the fixed points), A335879.

%Y Cf. A000043, A000668, A000396, A324200.

%K nonn

%O 1,3

%A _Antti Karttunen_, Feb 09 2020