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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 20 01:14 EDT 2019. Contains 326136 sequences. (Running on oeis4.)