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A354823 Dirichlet inverse of A351083, where A351083(n) = gcd(n, A327860(n)), and A327860 is the arithmetic derivative of the primorial base exp-function.

%I A354823 #9 Jun 09 2022 11:41:24
%S A354823 1,-1,-1,-1,-1,1,-7,-5,0,1,-1,1,-1,13,-3,-1,-1,-2,-1,-7,13,1,-1,9,-24,
%T A354823 1,0,7,-1,7,-1,33,1,-15,9,-6,-1,1,-11,27,-1,-25,-1,-1,4,1,-1,7,48,24,
%U A354823 1,-1,-1,2,-3,59,1,1,-1,19,-1,1,-12,23,1,-1,-1,33,1,-23,-1,-2,-1,1,52,1,7,23,-1,-67,0,1,-1,-25
%N A354823 Dirichlet inverse of A351083, where A351083(n) = gcd(n, A327860(n)), and A327860 is the arithmetic derivative of the primorial base exp-function.
%H A354823 Antti Karttunen, <a href="/A354823/b354823.txt">Table of n, a(n) for n = 1..30030</a>
%H A354823 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A354823 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A351083(n/d) * a(d).
%o A354823 (PARI)
%o A354823 A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
%o A354823 A351083(n) = gcd(n, A327860(n));
%o A354823 memoA354823 = Map();
%o A354823 A354823(n) = if(1==n,1,my(v); if(mapisdefined(memoA354823,n,&v), v, v = -sumdiv(n,d,if(d<n,A351083(n/d)*A354823(d),0)); mapput(memoA354823,n,v); (v)));
%Y A354823 Cf. A327860, A351083.
%Y A354823 Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
%Y A354823 Cf. also A346242, A354347, A354348, A354824.
%K A354823 sign
%O A354823 1,7
%A A354823 _Antti Karttunen_, Jun 09 2022