A236852 Remultiply n first "downward", from N to GF(2)[X], and then remultiply that result back "upward", from GF(2)[X] to N: a(n) = A234742(A234741(n)).

0, 1, 2, 3, 4, 9, 6, 7, 8, 9, 18, 11, 12, 13, 14, 27, 16, 81, 18, 19, 36, 21, 22, 39, 24, 81, 26, 27, 28, 33, 54, 31, 32, 33, 162, 63, 36, 37, 38, 39, 72, 41, 42, 75, 44, 81, 78, 47, 48, 49, 162, 243, 52, 57, 54, 99, 56, 57, 66, 59, 108, 61, 62, 63, 64, 117, 66, 67, 324

Offset: 0

Author

Antti Karttunen, Feb 02 2014

Keywords

• nonn
• mult

Comments

This sequence appears to be completely multiplicative with a(p) = A234742(p) although neither A234741 or A234742 are even multiplicative. Terms tested up to n = 10^7. - Andrew Howroyd, Aug 01 2018

Yes, this is true. Consider for example n = p*q*r*r, where p, q, r are primes in N. Then A234741(n) = p1 x q1 x q2 x q3 x r1 x r2 x r1 x r2, where p1, q1, r1, ..., are the irreducible factors of p, q, r when factored over GF(2), and x stands for multiplication in ring GF(2)[X] (A048720). [Note that these irreducible factors are not necessarily primes in N, except p1 (= p), which must be a term of A091206. Also, A234741(p) = p for any prime p.] Next, a(n) = A234742(p1 x q1 x q2 x q3 x r1 x r2 x r1 x r2) = p1 * q1 * q2 * q3 * r1 * r2 * r1 * r2, which can be obtained also as a(p)*a(q)*a(r)*a(r), thus proving the multiplicativity. - Antti Karttunen, Aug 02 2018

Examples

			From _Antti Karttunen_, Aug 02 2018: (Start)
For n = 3, we have A234741(3) = 3 = 11 in binary, which encodes a (0,1)-polynomial x+1, which is irreducible over GF(2) thus A234742(3) = 3 and  a(3) = 3.
For n = 5, we have A234741(5) = 5 = 101 in binary, which encodes a (0,1)-polynomial x^2 + 1, which factorizes as (x+1)(x+1) when factored over GF(2), that is 5 = A048720(3,3), thus it follows that A234742(5) = 3*3 = 9, and a(5) = 9.
For n = 9 = 3*3, we have A234741(9) = A048720(3,3) = 5, and A234742(5) = 9 as shown above. Also by multiplicativity, we have a(3*3) = a(3)*a(3) = 3*3 = 9.
(End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Index entries for sequences related to polynomials in ring GF(2)[X]

Crossrefs

Fixed points: A236860.

Cf. A048720, A234741, A234742, A236851, A236836-A236837, A236846-A236847.

Programs

Scheme

(define (A236852 n) (A234742 (A234741 n)))

Formula

a(n) = A234742(A234741(n)).

Extensions

Keyword mult added after Andrew Howroyd's observation. - Antti Karttunen, Aug 02 2018