A236861 - OEIS

%I #12 May 15 2014 10:19:05

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,0,1,0,1,1,1

%N Characteristic function of A236841 (A234741): a(n) = 1 if n is a result of downward remultiplication (N -> GF(2)[X]) of some number, 0 otherwise.

%H Antti Karttunen, <a href="/A236861/b236861.txt">Table of n, a(n) for n = 0..8192</a>

%H A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program, containing a few functions needed for computing this sequence.</a>

%F a(0)=1, a(1)=1, a(p)=1 for primes p, and for composite n which are irreducible in GF(2)[X] (terms of A091214), a(n)=0, and for other composite n (terms of A091212), a(n) = 1 only if there exists at least one such pair of k, m < n that n = A048720(k,m) and a(k)*a(m) = 1, and 0 otherwise.

%F (Please compare also to the formula for A236862).

%o (Scheme, two quite inefficient implementations, which require also definec and/or GF2Xdivide from a linked Scheme-source file)

%o (definec (A236861 n) (if (zero? (A236833 n)) 0 1))

%o (definec (A236861 n) (cond ((< n 2) 1) ((prime? n) 1) ((= 1 (A091225 n)) 0) (else (let loop ((d 2)) (cond ((= d n) 0) ((and (= d (A091255bi n d)) (not (zero? (* (A236861 d) (A236861 (GF2Xdivide n d)))))) 1) (else (loop (+ d 1))))))))

%Y Positions of zeros occur in the same positions as in A236833; sequence A236834 gives these positions.

%Y Cf. A236841, A234741, A236862, A048720, A091225, A091255.

%K nonn

%O 0

%A _Antti Karttunen_, Apr 13 2014