A374113 - OEIS

%I #7 Jun 29 2024 11:17:53

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

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

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

%N a(n) = 1 if A113177(n) and A276085(n) are both even, otherwise 0, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively.

%C a(n) = 1 if the 2-adic valuation of n is even, and the number of its prime factors (with multiplicity, A001222) and its 3-adic valuation (A007949) have the same parity, otherwise 0.

%H Antti Karttunen, <a href="/A374113/b374113.txt">Table of n, a(n) for n = 1..100000</a>

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

%F a(n) = A035263(n) * A373585(n).

%F a(n) = A059841(A374112(n)).

%o (PARI)

%o A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));

%o A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };

%o A374113(n) = (!(A113177(n)%2) && !(A276085(n)%2));

%o (PARI) A374113(n) = (!(valuation(n,2)%2) && !((bigomega(n)-valuation(n, 3))%2));

%Y Characteristic function of A374114, whose complement A374115 gives the indices of 0's.

%Y Cf. A001222, A007949, A035263, A113177, A059841, A276085, A373585, A374112.

%K nonn

%O 1

%A _Antti Karttunen_, Jun 29 2024