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A300563 Condensed deep factorization of n, A300562(n) written in decimal: floor of odd part of A300561(n) divided by 2.

%I A300563 #12 Mar 17 2018 17:54:19
%S A300563 0,1,28,7,484,412,496,124,115,6628,7972,7708,7396,6640,117220,31,8068,
%T A300563 1651,8080,123364,117232,106276,7792,127516,1939,105700,1852,123376,
%U A300563 118564,1690084,129316,2020,1875748,106372,1985008,30835,127204,106384,1875172,2040292,124708
%N A300563 Condensed deep factorization of n, A300562(n) written in decimal: floor of odd part of A300561(n) divided by 2.
%C A300563 The binary representation of the deep factorization of n, A300560, is obtained by recursively replacing any factor prime(i)^e_i by the expression [i [e_i]], and finally taking '[' and ']' as binary digits 1 and 0.
%C A300563 This always ends in trailing 0's which can be safely removed without loss of information; then there is a final binary digit 1 that can also be dropped. The result is A300562(n) in binary, equal to a(n) when converted to decimal.
%C A300563 The initial a(1) = 0 results from the empty factorization of 1.
%C A300563 To reconstruct the deep factorization of n > 1, take a(n)*2+1, multiply by 2^A145037(a(n)*2+1) (i.e., number of bits = 1 minus number of bits = 0), and write it in binary.
%F A300563 a(n) = A004526(A000265(A300561(n))), where A004526 = floor(./2) and A000265(x) = x/2^A007814(x) is the odd part of x, A007814 is 2-adic valuation. - _M. F. Hasler_, Mar 16 2018
%e A300563 The first term a(1) = 0 represents, by convention, the empty factorization of the number 1.
%e A300563 The binary-coded deep factorization is restored as follows (and a(n) calculated from this going the opposite direction):
%e A300563 a(2) = 1, append a bit 1 or do 1 X 2 + 1 = 3 = 11[2]. This has 2 bits 1, no bit 0 so append 2 bits 0 => A300560(2) = 1100 in binary, or 12 = A300561(2) in decimal.
%e A300563 a(3) = 28 = 11100[2], append a bit 1 or do 28 X 2 + 1 = 57 = 111001[2]. This has 4 bits 1 and 2 bits 0, so append two more of the latter => A300560(3) = 11100100 in binary or A300561(3) = 228 in decimal.
%e A300563 a(4) = 7 = 111[2], append a bit 1 or do 7 X 2 + 1 = 15 = 1111[2]. This has 4 bits 1 and no bit 0 so append 4 0's => 11110000 = A300560(4) or A300561(4) = 240 in decimal.
%e A300563 See A300560 for conversion of this binary coding of the deep factorization into the ordinary factorization.
%o A300563 (PARI) A300563(n)=(n=A300561(n))>>(valuation(n+!n,2)+1)
%Y A300563 Cf. A300560, A300561, A300562.
%Y A300563 Cf. A061396, A062504, A062860; A004526, A007814.
%K A300563 nonn
%O A300563 1,3
%A A300563 _M. F. Hasler_, Mar 08 2018