This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349350 #15 Dec 24 2023 02:41:44
%S A349350 1,-1,-1,-3,-1,1,-1,-1,-8,1,-1,3,-1,1,1,5,-1,8,-1,3,1,1,-1,1,-24,1,
%T A349350 -10,3,-1,-1,-1,7,1,1,1,24,-1,1,1,1,-1,-1,-1,3,8,1,-1,-5,-48,24,1,3,
%U A349350 -1,10,1,1,1,1,-1,-3,-1,1,8,-3,1,-1,-1,3,1,-1,-1,8,-1,1,24,3,1,-1,-1,-5,28,1,-1,-3,1,1,1,1,-1,-8
%N A349350 Dirichlet inverse of A057521, the powerful part of n.
%C A349350 Multiplicative because A057521 is.
%H A349350 Antti Karttunen, <a href="/A349350/b349350.txt">Table of n, a(n) for n = 1..20000</a>
%F A349350 a(1) = 1; a(n) = -Sum_{d|n, d < n} A057521(n/d) * a(d).
%F A349350 Let p be a prime, B = 1 + p - 2*p^2 and C = sqrt(1 + 2*p - 3*p^2). Then the sequence is multiplicative with a(p^e) = ((B-C)*(p-1+C)^(e-1) - (B+C)*(p-1-C)^(e-1))/(2^e*C). - _Sebastian Karlsson_, Dec 02 2021
%t A349350 f[p_, e_] := Module[{B = 1 + p - 2*p^2, C = Sqrt[1 + 2*p - 3*p^2]}, FullSimplify[((B - C)*(p - 1 + C)^(e - 1) - (B + C)*(p - 1 - C)^(e - 1))/(2^e*C)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 24 2023 *)
%o A349350 (PARI)
%o A349350 A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
%o A349350 memoA349350 = Map();
%o A349350 A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(d<n,A057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));
%Y A349350 Cf. A057521.
%Y A349350 Cf. also A349340, A349442.
%K A349350 sign,mult
%O A349350 1,4
%A A349350 _Antti Karttunen_, Nov 18 2021