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Characteristic function of A001317.
5

%I #24 Jun 30 2022 14:43:26

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

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

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

%N Characteristic function of A001317.

%C a(k) = 1 iff k is in A001317, and 0 for all other values.

%C The recursive formula is based on the fact that only from the terms of A001317 we can reach all the way down to 1 when repeatedly applying the map k -> A006068(k)/2 as long as it is possible to iterate (before A006068(k) is odd).

%C This sequence is not multiplicative. The smallest counterexample is for n = A000215(6) = 4294967297 which is the first composite Fermat number. In this case a(n) = 1 which is not the product of a(641) and a(6700417) which are both zero. - _Andrew Howroyd_, Aug 08 2018

%H Antti Karttunen, <a href="/A268384/b268384.txt">Table of n, a(n) for n = 0..4369</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(0) = 0, a(1) = 1, and for n > 1, a(n) = 0 if A006068(n) is odd, otherwise a(A006068(n)/2).

%F a(n) = A209229(A193231(n)).

%o (Scheme, two variants)

%o (definec (A268384 n) (cond ((<= n 1) n) ((odd? (A006068 n)) 0) (else (A268384 (/ (A006068 n) 2))))) ;; Uses the memoization-macro definec

%o (define (A268384 n) (A209229 (A193231 n)))

%o (Python)

%o from itertools import count, islice

%o def A268384_gen(): # generator of terms

%o a = -1

%o for n in count(0):

%o b = int(''.join(str(int(not(~n&k))) for k in range(n+1)), 2)

%o yield from (0,)*(b-a-1)

%o yield 1

%o a = b

%o A268384_list = list(islice(A268384_gen(),30)) # _Chai Wah Wu_, Jun 30 2022

%Y Cf. A001317, A006068, A193231, A209229.

%Y Cf. also A000215, A268389, A268391.

%K nonn

%O 0

%A _Antti Karttunen_, Feb 10 2016