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A323893 Dirichlet inverse of A048673, where A048673(n) = (A003961(n)+1) / 2, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

%I A323893 #13 Nov 30 2024 12:54:51
%S A323893 1,-2,-3,-1,-4,4,-6,-2,-4,5,-7,3,-9,7,6,-4,-10,8,-12,4,8,8,-15,8,-9,
%T A323893 10,-12,6,-16,5,-19,-8,9,11,9,8,-21,13,11,11,-22,11,-24,7,16,16,-27,
%U A323893 20,-25,18,12,9,-30,32,10,17,14,17,-31,6,-34,20,24,-16,12,14,-36,10,17,20,-37,16,-40,22,27,12,12,20,-42,28,-36,23,-45,12,13
%N A323893 Dirichlet inverse of A048673, where A048673(n) = (A003961(n)+1) / 2, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).
%H A323893 Antti Karttunen, <a href="/A323893/b323893.txt">Table of n, a(n) for n = 1..16383</a>
%H A323893 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A323893 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A048673(n/d) * a(d).
%F A323893 a(n) = A349134(A003961(n)). - _Antti Karttunen_, Nov 30 2024
%o A323893 (PARI)
%o A323893 up_to = 20000;
%o A323893 DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A323893 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A323893 A048673(n) = (A003961(n)+1)/2;
%o A323893 v323893 = DirInverse(vector(up_to,n,A048673(n)));
%o A323893 A323893(n) = v323893[n];
%o A323893 (PARI)
%o A323893 memoA323893 = Map();
%o A323893 A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(d<n,A048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v))); \\ _Antti Karttunen_, Nov 30 2024
%Y A323893 Cf. A003961, A048673, A323894, A349134, A378520 (Möbius transform).
%K A323893 sign
%O A323893 1,2
%A A323893 _Antti Karttunen_, Feb 08 2019