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A378606 Dirichlet convolution of A046692 and A003961, where A046692 is the Dirichlet inverse of sigma, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

%I A378606 #18 Dec 12 2024 09:30:01
%S A378606 1,0,1,2,1,0,3,6,8,0,1,2,3,0,1,18,1,0,3,2,3,0,5,6,12,0,40,6,1,0,5,54,
%T A378606 1,0,3,16,3,0,3,6,1,0,3,2,8,0,5,18,40,0,1,6,5,0,1,18,3,0,1,2,5,0,24,
%U A378606 162,3,0,3,2,5,0,1,48,5,0,12,6,3,0,3,18,200,0,5,6,1,0,1,6,7,0,9,10,5,0,3,54,3,0,8,24
%N A378606 Dirichlet convolution of A046692 and A003961, where A046692 is the Dirichlet inverse of sigma, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).
%H A378606 Antti Karttunen, <a href="/A378606/b378606.txt">Table of n, a(n) for n = 1..20000</a>
%H A378606 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%H A378606 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A378606 a(n) = Sum_{d|n} A046692(d)*A003961(n/d).
%F A378606 a(n) = Sum_{d|n} A008683(d)*A349387(n/d).
%F A378606 Multiplicative with a(p^e) = q(p)^e - (p+1) * q(p)^(e-1) + p * q(p)^(e-2) if e >= 2, and q(p) - p  - 1 if e = 1, where q(p) = A151800(p) is the prime next to p. - _Amiram Eldar_, Dec 11 2024
%t A378606 f[p_, e_] := Module[{q = NextPrime[p]}, If[e == 1, q - p - 1, q^e - (p + 1)*q^(e - 1) + p*q^(e - 2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 11 2024 *)
%o A378606 (PARI)
%o A378606 A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A378606 A046692(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(f[1,i]+1), if(2==f[2,i], f[1,i], 0))); };
%o A378606 A378606(n) = sumdiv(n,d,A046692(d)*A003961(n/d));
%Y A378606 Cf. A003961, A008683, A016825 (positions of 0's), A046692, A151800, A349387 (inverse Möbius transform), A378607 (Dirichlet inverse).
%K A378606 nonn,mult
%O A378606 1,4
%A A378606 _Antti Karttunen_, Dec 11 2024