A346234 - OEIS

%I #19 May 25 2022 08:42:29

%S 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 0,51,0,0,-31,-105,-37,0,65,57,77,0,-41,69,85,0,-43,-165,-47,0,0,87,

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

%H Antti Karttunen, <a href="/A346234/b346234.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = A055615(A003961(n)).

%F a(n) = A008683(n) * A003961(n).

%F Multiplicative with a(p^e) = 0 if e > 1, and -nextprime(p) otherwise, where nextprime function is A151800. - _Antti Karttunen_, Nov 14 2021

%o (PARI)

%o up_to = 16384;

%o 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 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o v346234 = DirInverseCorrect(vector(up_to,n,A003961(n)));

%o A346234(n) = v346234[n];

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A346234(n) = (moebius(n)*A003961(n));

%o (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 Cf. A003961, A008683, A055615, A151800.

%Y Cf. also A046692, A323893, A349125.

%K sign,mult

%O 1,2

%A _Antti Karttunen_, Jul 11 2021

%E Keyword:mult added by _Antti Karttunen_, Nov 14 2021