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A378520 Dirichlet inverse of A336840, where A336840 is the inverse Möbius transform of A048673.

%I A378520 #10 Mar 31 2025 01:47:40
%S A378520 1,-3,-4,1,-5,10,-7,-1,-1,12,-8,-2,-10,16,14,-2,-11,5,-13,-2,18,18,
%T A378520 -16,6,-5,22,-8,-2,-17,-20,-20,-4,20,24,20,1,-22,28,24,8,-23,-20,-25,
%U A378520 -2,11,34,-28,14,-19,18,26,-2,-31,32,22,12,30,36,-32,4,-35,42,17,-8,26,-20,-37,-2,36,-14,-38,3,-41,46,26,-2,26
%N A378520 Dirichlet inverse of A336840, where A336840 is the inverse Möbius transform of A048673.
%H A378520 Antti Karttunen, <a href="/A378520/b378520.txt">Table of n, a(n) for n = 1..20000</a>
%H A378520 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%F A378520 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A336840(n/d) * a(d).
%F A378520 a(n) = Sum_{d|n} A008683(n/d)*A323893(d).
%F A378520 a(n) = A349915(A003961(n)).
%o A378520 (PARI)
%o A378520 A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
%o A378520 A336840(n) = sumdiv(n,d,A048673(d));
%o A378520 memoA378520 = Map();
%o A378520 A378520(n) = if(1==n,1,my(v); if(mapisdefined(memoA378520,n,&v), v, v = -sumdiv(n,d,if(d<n,A336840(n/d)*A378520(d),0)); mapput(memoA378520,n,v); (v)));
%Y A378520 Möbius transform of A323893.
%Y A378520 Dirichlet inverse of A336840.
%Y A378520 Cf. A003961, A008683, A349915.
%K A378520 sign
%O A378520 1,2
%A A378520 _Antti Karttunen_, Nov 30 2024