A300560 Deep factorization of n, written in binary: replace each factor p^e with the expression [primepi(p) [ e ]], iterate on these numbers, finally replace '[' and ']' with '1' and '0'.

0, 1100, 11100100, 11110000, 111100100100, 110011100100, 111110000100, 111110010000, 111001110000, 1100111100100100, 1111100100100100, 1111000011100100, 1110011100100100, 1100111110000100, 11100100111100100100, 111111000000, 1111110000100100, 1100111001110000, 1111110010000100

Offset

1,2

Comments

Consider the prime factorization of n, replace each factor prime(i)^e_i with the parenthesized expression [i [e_i]], iterate this process on the indices i and exponents e_i, and finally replace '[' and ']' with digits '1' and '0'.

See A300561 for the decimal representation of these binary numbers.

There is redundancy: trailing '0's can be removed without loss of information; then each term ends in a digit 1 which can also be removed. This more condensed version is given in A300562, the decimal representation in A300563(n) = (m/2^valuation(m,2) - 1)/2 with m = a(n) [read in binary] = A300561(n).

The initial a(1) = 0 represents the empty string of binary digits.

Links

Table of n, a(n) for n=1..19.

J. Awbrey, Riffs and Rotes, Selected Sequences, OEIS wiki, Feb. 2010.

Example

The first term a(1) = 0 represents, by convention, the empty factorization of the number 1.
2 = prime(1)^1 => (1(1)) => (()) => 1100 = a(2). (The 1's disappear, having empty factorization.)
3 = prime(2)^1 => (2(1)) => ((())()) [using 2 => (())] => 11100100 = a(3).
4 = prime(1)^2 => (1(2)) => (((()))) => 11110000 = a(4).
5 = prime(3)^1 => (3(1)) => (((())())()) => 111100100100 = a(5).
6 = prime(1)^1*prime(2)^1 => (1(1))(2(1)) => (())((())()) => 110011100100 = a(6) (= concatenation of a(2) and a(3), since 6 = 2*3.)
7 = prime(4)^1 => (4(1)) => ((((())))()) => 111110000100 = a(7).
8 = prime(1)^3 => (1(3)) => ((((())()))) => 111110010000 = a(8), and so on.
To convert back to the usual factorization, replace 0 and 1 by ')' and '(', then iteratively replace any (x(y)) by prime_x^y, where an empty x or y means 1.
Examples: 1100 = (()) = (x(y)) with x = y = 1, so (()) = prime_1^1 = 2.
110011100100 = _(())_(_(())_()) = 2 (2()) = 2 prime_2^1 = 6.
111110010000 = (((_(())_()))) = ((_(2())_)) = ((3)) = prime_1^3 = 8.

Prog

(PARI) A300560(n)=(n=factor(n))||return(""); n[, 1]=apply(primepi, n[, 1]); concat(apply(t->Str("1"t[1]"1"t[2]"00"), Col(apply(A300560, n))~))

Crossrefs

Cf. A300561, A300562, A300563.

Cf. A061396, A062504, A062860.

Sequence in context: A050926 A083933 A080317 * A068279 A278343 A277864

Adjacent sequences: A300557 A300558 A300559 * A300561 A300562 A300563

Keyword

nonn

Author

M. F. Hasler, Mar 08 2018

Status

approved