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A300561 Deep factorization of n, A300560, converted from binary to decimal. (Binary digits obtained by recursively replacing each factor p^e with [primepi(p) [e]], then '[' = 1, ']' = 0.) 4
0, 12, 228, 240, 3876, 3300, 3972, 3984, 3696, 53028, 63780, 61668, 59172, 53124, 937764, 4032, 64548, 52848, 64644, 986916, 937860, 850212, 62340, 1020132, 62064, 845604, 59280, 987012, 948516, 13520676, 1034532, 64656, 15005988, 850980, 15880068, 986736, 1017636 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Convert to decimal the binary numbers A300560, which represent the deep factorization of n: each factor prime(i)^e_i is replaced by the expression [i [e_i]], recursively for indices i and exponents e_i, and finally '[' and ']' are considered as binary digits 1 and 0.

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

All terms are multiples of 4, and some of a higher power of 2, which represent the trailing closing parentheses of the deep factorization. These factors of 2 can be removed without loss of information; then all terms (except for n = 1) are odd, and we can consider (x-1)/2. This more condensed version is A300563(n) = (a(n)/2^valuation(a(n),2) - 1)/2, with binary representation given in A300562(n).

LINKS

Table of n, a(n) for n=1..37.

J. Awbrey, https://oeis.org/wiki/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_2 = 12 = a(2).

3 = prime(2)^1 => (2(1)) => ((())()) => 11100100_2 = 228 = a(3).

4 = prime(1)^2 => (1(2)) => (((()))) => 11110000_2 = 240 = a(4).

5 = prime(3)^1 => (3(1)) => (((())())()) => 111100100100_2 = 3876 = a(5).

6 = prime(1)^1*prime(2)^1 => (1(1))(2(1)) => (())((())()) => 110011100100_2 = 3300 = a(6).

7 = prime(4)^1 => (4(1)) => ((((())))()) => 111110000100_2 = 3972 = a(7).

8 = prime(1)^3 => (1(3)) => ((((())()))) => 111110010000_2 = 3984 = a(8), and so on.

PROG

(PARI) A300561(n)=fromdigits(digits(eval(A300560(n))), 2)

CROSSREFS

Cf. A300560, A300562, A300563.

Cf. A061396, A062504, A062860.

Sequence in context: A220068 A247745 A098647 * A290754 A318417 A034832

Adjacent sequences:  A300558 A300559 A300560 * A300562 A300563 A300564

KEYWORD

nonn

AUTHOR

M. F. Hasler, Mar 08 2018

STATUS

approved

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Last modified June 5 01:27 EDT 2020. Contains 334828 sequences. (Running on oeis4.)