A235145 - OEIS

%I #20 Aug 06 2017 03:20:04

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

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

%U 0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,2

%N a(n) = Number of steps to reach a fixed point or 2-cycle, when iterating A235027 starting from value n.

%C Equally, a(n) = minimum number of steps needed to repeat k = A235027(k)(starting from k = n) until A001222(A235027(k)) = A001222(k).

%C Or in other words, how many times are needed to repeatedly factorize the number, to reverse the bits of each odd prime factor (with A056539) and factorize and bit-reverse the reversed factors again, until the number of prime divisors no more grows, meaning that we have found either a fixed point or entered a cycle of two.

%H Antti Karttunen, <a href="/A235145/b235145.txt">Table of n, a(n) for n = 0..10001</a>

%F If A235027(A235027(n)) = n, a(n)=0, otherwise 1+a(A235027(n)).

%F Equally, if A001222(A235027(n)) = A001222(n), a(n)=0, otherwise 1+a(A235027(n)).

%F a(2n) = a(n), and in general, for composite values a(u * v) = max(a(u),a(v)).

%F For composite n, a(n) = Max_{p|n} a(p). [The above reduces to this: select the maximal value from all values a(p) computed for primes p dividing n]

%F For prime p, a(p) = 0 if A056539(p) is also prime (p is 2 or in A074832), otherwise a(p) = 1+a(A056539(p)).

%e 19, '10011' in binary, when reversed, yields '11001' = 25, when factored, yields 5 * 5, ('101' * '101' in binary), which divisors stay same when reversed, thus it took one iteration step to reach a point where the number of prime divisors no more grows. Thus a(19)=1.

%o (Scheme, two alternative definitions using memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)

%o (definec (A235145 n) (cond ((= (A235027 (A235027 n)) n) 0) (else (+ 1 (A235145 (A235027 n))))))

%o (definec (A235145 n) (cond ((= (A001222 (A235027 n)) (A001222 n)) 0) (else (+ 1 (A235145 (A235027 n))))))

%o (PARI) revbits(n) = fromdigits(Vecrev(binary(n)), 2);

%o a235027(n) = {f = factor(n); for (k=1, #f~, if (f[k,1] != 2, f[k,1] = revbits(f[k,1]););); factorback(f);}

%o find(v, newn) = {for (k=1, #v, if (v[#v -k + 1] == newn, return (k));); return (0);}

%o a(n) = {ok = 0; v = [n]; while (! ok, newn = a235027(n); ind = find(v, newn); if (ind, ok = 1, v = concat(v, newn); n = newn);); #v - ind;} \\ _Michel Marcus_, Aug 06 2017

%Y A235146 gives the positions of records. Cf. A001222, A056539, A074832, A235027.

%K nonn,base

%O 0,140

%A _Antti Karttunen_, Jan 03 2014