

A289271


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


9



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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

(PARI) See Links section.
(PARI) A289271(n) = { my(f = factor(n), pps = vecsort(vector(#f~, i, f[i, 1]^f[i, 2])), s=0, x=1, pp=1, k=1); for(i=1, #f~, while(pp < pps[i], pp++; while(!isprimepower(pp)(gcd(pp, x)>1), pp++); k++); s += 2^k; x *= pp); (s); }; \\ Antti Karttunen, Jan 01 2019


CROSSREFS

Cf. A000120, A000961, A001221, A002110, A007814, A034684, A034729, A087207, A289272 (inverse), A322988, A322990, A322991, A322992, A322995.
Cf. also A156552, A052331 for similar constructions.
Sequence in context: A317503 A243505 A243065 * A223699 A231610 A225124
Adjacent sequences: A289268 A289269 A289270 * A289272 A289273 A289274


KEYWORD

nonn,base


AUTHOR

Rémy Sigrist, Jun 30 2017


STATUS

approved



