A332215 Mersenne-prime fixing variant of A243071: a(n) = A243071(A332213(n)).

0, 1, 3, 2, 15, 6, 7, 4, 5, 30, 63, 12, 255, 14, 29, 8, 511, 10, 1023, 60, 13, 126, 2047, 24, 23, 510, 9, 28, 4095, 58, 31, 16, 125, 1022, 27, 20, 16383, 2046, 509, 120, 32767, 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

Offset: 1

Author

Antti Karttunen, Feb 09 2020

Keywords

• nonn

Comments

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.

From Antti Karttunen, Jul 10 2020: (Start)

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

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.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

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

Cf. A000043, A000668, A000396, A324200.

Programs

PARI

A332215(n) = A243071(A332213(n));

Formula

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

For all n >= 1, a(A335431(n)) = A335431(n), a(A335882(n)) = A335879(n). - Antti Karttunen, Jul 10 2020