A289271 A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition).

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

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^n-1 for any n >= 0.

a(A000961(n+1)) = 2^(n-1) for any n > 0.

A000120(a(n)) = A001221(n) for any n > 0 (each prime divisor p of n (alongside the p-adic valuation of n) is encoded as a single 1 bit in the base-2 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: A243505 A243065 A352782 * A354169 A341811 A332306

Adjacent sequences: A289268 A289269 A289270 * A289272 A289273 A289274

Keyword

nonn,base

Author

Rémy Sigrist, Jun 30 2017

Status

approved