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

0, 1, 28, 7, 484, 412, 496, 124, 115, 6628, 7972, 7708, 7396, 6640, 117220, 31, 8068, 1651, 8080, 123364, 117232, 106276, 7792, 127516, 1939, 105700, 1852, 123376, 118564, 1690084, 129316, 2020, 1875748, 106372, 1985008, 30835, 127204, 106384, 1875172, 2040292, 124708

Offset: 1

Author

M. F. Hasler, Mar 08 2018

Keywords

• nonn

Comments

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.

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.

The initial a(1) = 0 results from the empty factorization of 1.

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.

Examples

			The first term a(1) = 0 represents, by convention, the empty factorization of the number 1.
The binary-coded deep factorization is restored as follows (and a(n) calculated from this going the opposite direction):
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.
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.
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.
See A300560 for conversion of this binary coding of the deep factorization into the ordinary factorization.
		

Crossrefs

Cf. A300560, A300561, A300562.

Cf. A061396, A062504, A062860; A004526, A007814.

Programs

PARI

A300563(n)=(n=A300561(n))>>(valuation(n+!n,2)+1)

Formula

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