cp's OEIS Frontend

A347096 a(1) = 1; a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d), where A341512(n) = sigma(n)*A003961(n) - n*sigma(A003961(n)).

%I A347096 #11 Nov 15 2021 03:39:57
%S A347096 1,-1,-2,-10,-2,-32,-4,-64,-42,-54,-2,-214,-4,-112,-112,-316,-2,-469,
%T A347096 -4,-412,-232,-168,-6,-792,-90,-262,-612,-860,-2,-1208,-6,-1216,-340,
%U A347096 -354,-320,-1655,-4,-484,-532,-1760,-2,-2528,-4,-1438,-1850,-732,-6,160,-364,-1863,-712,-2210,-6,-4596,-384,-3696,-976,-942,-2
%N A347096 a(1) = 1; a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d), where A341512(n) = sigma(n)*A003961(n) - n*sigma(A003961(n)).
%C A347096 Dirichlet inverse of the pointwise sum of A341512 and A063524 (1, 0, 0, 0, ...).
%H A347096 Antti Karttunen, <a href="/A347096/b347096.txt">Table of n, a(n) for n = 1..10000</a>
%H A347096 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%H A347096 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A347096 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A347096 a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d).
%F A347096 For all n >= 1, a(A000040(n)) = -A001223(n).
%o A347096 (PARI)
%o A347096 up_to = 16384;
%o A347096 A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A347096 A341512(n) = { my(u=A003961(n)); ((sigma(n)*u) - (n*sigma(u))); };
%o A347096 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 A347096 Aux347096(n) = if(1==n,n,A341512(n));
%o A347096 v347096 = DirInverseCorrect(vector(up_to,n,Aux347096(n)));
%o A347096 A347096(n) = v347096[n];
%Y A347096 Cf. A000203, A001223, A003961, A063524, A341512, A347097.
%Y A347096 Cf. also A346235, A346239.
%K A347096 sign,look
%O A347096 1,3
%A A347096 _Antti Karttunen_, Aug 19 2021