A359589 - OEIS

%I #16 Jan 11 2023 15:58:56

%S 1,0,0,0,0,-4,0,-2,-5,0,0,0,0,-2,-1,0,0,0,0,-2,-9,0,0,0,-9,-14,-2,0,0,

%T 0,0,0,-13,0,-5,12,0,-20,-1,0,0,0,0,-2,-2,-24,0,10,-13,-14,-9,-6,0,40,

%U -1,0,-1,0,0,0,0,-2,-2,2,-17,0,0,-2,-1,0,0,20,0,-2,-4,-4,-17,0,0,0,20,0,0,16,-1,-14,-1,-34

%N Dirichlet inverse of function f(n) = (-1 + gcd(A003415(n), A276086(n))), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

%H Antti Karttunen, <a href="/A359589/b359589.txt">Table of n, a(n) for n = 1..30030</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (A327858(n/d)-1) * a(d).

%F a(n) == A358777(n) mod 2.

%o (PARI)

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

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A327858(n) = gcd(A003415(n), A276086(n));

%o memoA359589 = Map();

%o A359589(n) = if(1==n,1,my(v); if(mapisdefined(memoA359589,n,&v), v, v = -sumdiv(n,d,if(d<n,(A327858(n/d)-1)*A359589(d),0)); mapput(memoA359589,n,v); (v)));

%Y Cf. A003415, A276086, A327858, A359595 (parity of terms), A359596 (positions of odd terms).

%Y Agrees paritywise with A358777.

%Y Cf. also A346242, A354347, A354348, A359603.

%K sign

%O 1,6

%A _Antti Karttunen_, Jan 09 2023