

A289271


A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition).


2



0, 1, 2, 4, 8, 3, 16, 32, 64, 5, 128, 6, 256, 9, 10, 512, 1024, 17, 2048, 12, 18, 33, 4096, 34, 8192, 65, 16384, 20, 32768, 7, 65536, 131072, 66, 129, 24, 36, 262144, 257, 130, 40, 524288, 11, 1048576, 68, 72, 513, 2097152, 258, 4194304, 1025, 514, 132
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OFFSET

1,3


COMMENTS

For n > 0, with prime factorization Product_{i=1..k} p_i ^ e_i (all p_i distinct and all e_i > 0):
 let S_n = A000961 \ { p_i ^ (e_i + j) with i=1..k and j > 0 },
 a(n) = Sum_{i=1..k} 2^#{ s in S_n with 1 < s < p_i ^ e_i }.
In an informal way, we encode the prime powers > 1 that are unitary divisors of n as 1's in binary, while discarding the 0's corresponding to their "proper" multiples.
a(A002110(n)) = 2^n1 for any n >= 0.
a(A000961(n+1)) = 2^(n1) for any n > 0.
A000120(a(n)) = A001221(n) for any n > 0 (each prime divisor p of n (alongside the padic valuation of n) is encoded as a single 1 bit in the base2 representation of a(n)).
A000961(2+A007814(a(n))) = A034684(n) for any n > 1 (the least significant bit of a(n) encodes the smallest unitary divisor of n that is larger than 1).
This sequence establishes a bijection between the positive numbers and the nonnegative numbers; see A289272 for the inverse of this sequence.
The numbers 4, 36, 40 and 532 equal their image; are there other such numbers?
This sequence has connections with A034729 (which encodes the divisors of a number, and is not surjective) and A087207 (which encodes the prime divisors of a number, and is not injective).


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, PARI program for A289271
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers


EXAMPLE

For n = 204 = 2^2 * 3 * 17:
 S_204 = A000961 \ { 2^3, 2^4, ..., 3^2, ... }
= { 1, 2, 3, 4, 5, 7, 11, 13, 17, ... },
 a(204) = 2^#{ 2, 3 } + 2^#{ 2 } + 2^#{ 2, 3, 4, 5, 7, 11, 13 }
= 2^2 + 2^1 + 2^7
= 134.
See also the illustration of the first terms in Links section.


PROG

See Links section.


CROSSREFS

Cf. A000120, A000961, A001221, A002110, A007814, A034684, A034729, A087207, A289272 (inverse).
Sequence in context: A277695 A243505 A243065 * A223699 A231610 A225124
Adjacent sequences: A289268 A289269 A289270 * A289272 A289273 A289274


KEYWORD

nonn,base


AUTHOR

Rémy Sigrist, Jun 30 2017


STATUS

approved



