login
a(n) = A353459(n) mod 2.
3

%I #12 Jan 06 2023 10:51:29

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

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

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

%N a(n) = A353459(n) mod 2.

%H Antti Karttunen, <a href="/A359469/b359469.txt">Table of n, a(n) for n = 1..100000</a>

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

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F For all n >= 1, a(n) = a(A003961(n)) = a(A348717(n)).

%o (PARI)

%o A353457(n) = { my(f=factor(n)); prod(i=1,#f~,if(!(primepi(f[i,1])%2), 1, if(f[i,2]==1, -1, 0))); };

%o A353458(n) = { my(f=factor(n)); prod(i=1,#f~,if(primepi(f[i,1])%2, 1, if(f[i,2]==1, -1, 0))); };

%o A353459(n) = (A353457(n)+A353458(n));

%o A359469(n) = (A353459(n)%2);

%o (Python)

%o from functools import reduce

%o from operator import iand

%o from sympy import factorint, primepi

%o def A359469(n):

%o f = [(primepi(p)&1, int(e==1)) for p, e in factorint(n).items()]

%o return reduce(iand,(e for p, e in f if not p),1)^reduce(iand,(e for p, e in f if p),1) # _Chai Wah Wu_, Jan 06 2023

%Y Characteristic function of A359470.

%Y Cf. A003961, A348717, A353457, A353458, A353459.

%Y Differs from A359466 and A359467 for the first time at n=100, as here a(100) = 1.

%K nonn

%O 1

%A _Antti Karttunen_, Jan 04 2023