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

1,3

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.

Links

Table of n, a(n) for n=1..41.

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

Example

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.

Prog

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

Crossrefs

Cf. A300560, A300561, A300562.

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

Sequence in context: A040764 A040763 A077752 * A040762 A040761 A070659

Adjacent sequences: A300560 A300561 A300562 * A300564 A300565 A300566

Keyword

nonn

Author

M. F. Hasler, Mar 08 2018

Status

approved