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A346234 Dirichlet inverse of A003961, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1).

%I A346234 #19 May 25 2022 08:42:29
%S A346234 1,-3,-5,0,-7,15,-11,0,0,21,-13,0,-17,33,35,0,-19,0,-23,0,55,39,-29,0,
%T A346234 0,51,0,0,-31,-105,-37,0,65,57,77,0,-41,69,85,0,-43,-165,-47,0,0,87,
%U A346234 -53,0,0,0,95,0,-59,0,91,0,115,93,-61,0,-67,111,0,0,119,-195,-71,0,145,-231,-73,0,-79,123,0,0,143,-255,-83,0,0,129
%N A346234 Dirichlet inverse of A003961, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1).
%H A346234 Antti Karttunen, <a href="/A346234/b346234.txt">Table of n, a(n) for n = 1..20000</a>
%H A346234 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A346234 a(n) = A055615(A003961(n)).
%F A346234 a(n) = A008683(n) * A003961(n).
%F A346234 Multiplicative with a(p^e) = 0 if e > 1, and -nextprime(p) otherwise, where nextprime function is A151800. - _Antti Karttunen_, Nov 14 2021
%o A346234 (PARI)
%o A346234 up_to = 16384;
%o A346234 DirInverseCorrect(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 A346234 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A346234 v346234 = DirInverseCorrect(vector(up_to,n,A003961(n)));
%o A346234 A346234(n) = v346234[n];
%o A346234 (PARI)
%o A346234 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A346234 A346234(n) = (moebius(n)*A003961(n));
%o A346234 (PARI) A346234(n) = { my(f = factor(n)); prod(i=1, #f~, if(1<f[i,2], 0, -nextprime(1+f[i,1]))); }; \\ _Antti Karttunen_, Nov 14 2021
%Y A346234 Cf. A003961, A008683, A055615, A151800.
%Y A346234 Cf. also A046692, A323893, A349125.
%K A346234 sign,mult
%O A346234 1,2
%A A346234 _Antti Karttunen_, Jul 11 2021
%E A346234 Keyword:mult added by _Antti Karttunen_, Nov 14 2021