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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.)

%I A300561 #19 Aug 11 2021 05:47:27
%S A300561 0,12,228,240,3876,3300,3972,3984,3696,53028,63780,61668,59172,53124,
%T A300561 937764,4032,64548,52848,64644,986916,937860,850212,62340,1020132,
%U A300561 62064,845604,59280,987012,948516,13520676,1034532,64656,15005988,850980,15880068,986736,1017636
%N 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.)
%C A300561 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.
%C A300561 The initial a(1) = 0 represents the empty string of binary digits.
%C A300561 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).
%H A300561 J. Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes#Selected_Sequences">Riffs and Rotes, Selected Sequences</a>, OEIS Wiki, Feb. 2010.
%e A300561 The first term a(1) = 0 represents, by convention, the empty factorization of the number 1.
%e A300561 2 = prime(1)^1 => (1(1)) => (()) => 1100_2 = 12 = a(2).
%e A300561 3 = prime(2)^1 => (2(1)) => ((())()) => 11100100_2 = 228 = a(3).
%e A300561 4 = prime(1)^2 => (1(2)) => (((()))) => 11110000_2 = 240 = a(4).
%e A300561 5 = prime(3)^1 => (3(1)) => (((())())()) => 111100100100_2 = 3876 = a(5).
%e A300561 6 = prime(1)^1*prime(2)^1 => (1(1))(2(1)) => (())((())()) => 110011100100_2 = 3300 = a(6).
%e A300561 7 = prime(4)^1 => (4(1)) => ((((())))()) => 111110000100_2 = 3972 = a(7).
%e A300561 8 = prime(1)^3 => (1(3)) => ((((())()))) => 111110010000_2 = 3984 = a(8), and so on.
%o A300561 (PARI) A300561(n)=fromdigits(digits(eval(A300560(n))),2)
%Y A300561 Cf. A300560, A300562, A300563.
%Y A300561 Cf. A061396, A062504, A062860.
%K A300561 nonn
%O A300561 1,2
%A A300561 _M. F. Hasler_, Mar 08 2018