A347098 - OEIS

%I #13 Nov 27 2021 11:03:44

%S 1,-1,-2,-4,-2,-5,-4,-10,-12,-7,-2,-1,-4,-11,-12,-16,-2,-1,-4,-7,-18,

%T -13,-6,42,-20,-17,-42,-5,-2,21,-6,-4,-24,-19,-26,106,-4,-23,-30,38,

%U -2,45,-4,-25,-10,-29,-6,196,-56,-17,-36,-23,-6,123,-28,82,-42,-31,-2,225,-6,-37,4,80,-38,15,-4,-43,-52,39,-2,413

%N a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

%C Dirichlet inverse of the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...).

%H Antti Karttunen, <a href="/A347098/b347098.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

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

%F a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d).

%F For all n >= 1, a(A000040(n)) = -A001223(n).

%t f[p_, e_] := NextPrime[p]^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Nov 27 2021 *)

%o (PARI)

%o up_to = 16384;

%o A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };

%o Aux347098(n) = if(1==n,n,A336853(n));

%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 v347098 = DirInverseCorrect(vector(up_to,n,Aux347098(n)));

%o A347098(n) = v347098[n];

%Y Cf. A000040, A001223, A003961, A336853, A347099, A347100.

%Y Cf. also A346248, A347096.

%K sign

%O 1,3

%A _Antti Karttunen_, Aug 19 2021