A075175 Prime factorization of n encoded by interleaving successive prime exponents in unary to bit-positions given by columns of A001477.

0, 1, 2, 5, 8, 3, 64, 37, 18, 9, 1024, 7, 32768, 65, 10, 549, 2097152, 19, 268435456, 13, 66, 1025, 68719476736, 39, 136, 32769, 274, 69, 35184372088832, 11, 36028797018963968, 16933, 1026, 2097153, 72, 23, 73786976294838206464

Offset: 1

Author

Antti Karttunen, Sep 13 2002

Keywords

• nonn

Comments

Here we store the exponent e_i of p_i (p1=2, p2=3, p3=5, ...) in the factorization of n to the bit positions given by the column i-1 of A001477 viewed as a table (the exponent of 2 is thus stored to bit positions 0, 2, 5, 9, 14, 20, ..., exponent of 3 to 1, 4, 8, 13, 19, ..., exponent of 5 to 3, 7, 12, 18, 25, ...) using unary system, i.e. we actually store 2^(e_i) - 1 in binary.

This injective mapping from N to N offers an example of the proof given in Cameron's book that any distributive lattice can be represented as a sublattice of the power-set lattice P(X) of some set X. With this we can implement GCD (A003989) with bitwise AND (A004198) and LCM (A003990) with bitwise OR (A003986). Also, to test whether x divides y, it is enough to check that ((a(x) OR a(y)) XOR a(y)) = A003987(A003986(a(x),a(y)),a(y)) is zero.

Examples

			a(24) = 39 because 24 = 2^3 * 3^1 so we add the binary words 100101 and 10 to get 100111 in binary = 39 in decimal and a(25) = 136 because 25 = 5^2 so we form a binary word 10001000 = 136 in decimal.
		

References

• P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1998, page 191. (12.3. Distributive lattices)

Crossrefs

Variant: A075173. Inverse: A075176.

A003989(x, y) = A075176(A004198(a(x), a(y))), A003990(x, y) = A075176(A003986(a(x), a(y))).