A373374 - OEIS

%I #15 Jun 03 2024 15:40:14

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

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

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

%N a(n) = 1 if both A001414(n) and A003415(n) are even, otherwise 0, where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

%H Antti Karttunen, <a href="/A373374/b373374.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) = A059841(A373364(n)).

%F a(n) = A356163(n) * A358680(n).

%F a(n) = A353374(n) + A253513(n)*A353374(n/8). [With shortcut + and *]

%o (PARI)

%o A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));

%o A373374(n) = (A353374(n) || (!(n%8) && A353374(n/8)));

%o (PARI)

%o A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A373374(n) = !(gcd(A001414(n), A003415(n))%2);

%Y Characteristic function of A373375, whose complement A373376 gives the positions of 0's.

%Y Positions of even terms in A373364.

%Y Cf. A001414, A003415, A059841, A253513, A353374, A356163, A358680.

%K nonn

%O 1

%A _Antti Karttunen_, Jun 03 2024