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A349387 Dirichlet convolution of A003961 with A055615 (Dirichlet inverse of n), where A003961 is fully multiplicative with a(p) = nextprime(p).

%I A349387 #21 Dec 11 2024 20:17:48
%S A349387 1,1,2,3,2,2,4,9,10,2,2,6,4,4,4,27,2,10,4,6,8,2,6,18,14,4,50,12,2,4,6,
%T A349387 81,4,2,8,30,4,4,8,18,2,8,4,6,20,6,6,54,44,14,4,12,6,50,4,36,8,2,2,12,
%U A349387 6,6,40,243,8,4,4,6,12,8,2,90,6,4,28,12,8,8,4,54,250,2,6,24,4,4,4,18,8,20,16,18,12,6
%N A349387 Dirichlet convolution of A003961 with A055615 (Dirichlet inverse of n), where A003961 is fully multiplicative with a(p) = nextprime(p).
%C A349387 Multiplicative because A003961 and A055615 are.
%C A349387 Convolving this with A000010 gives A003972, and convolving this with A000203 gives A003973.
%C A349387 Multiplicative with a(p^e) = nextprime(p)^e - p * nextprime(p)^(e-1), where nextprime function is A151800. - _Amiram Eldar_, Nov 18 2021
%H A349387 Antti Karttunen, <a href="/A349387/b349387.txt">Table of n, a(n) for n = 1..20000</a>
%H A349387 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%H A349387 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349387 a(n) = Sum_{d|n} A003961(n/d) * A055615(d).
%F A349387 For all n >= 1, a(A000040(n)) = A001223(n).
%t A349387 f[p_,e_] := (q = NextPrime[p])^e - p * q^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349387 (PARI)
%o A349387 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349387 A055615(n) = (n*moebius(n));
%o A349387 A349387(n) = sumdiv(n,d,A003961(n/d)*A055615(d));
%Y A349387 Cf. A000040, A001223, A003961, A055615, A151800, A349388 (Dirichlet inverse), A349389 (sum with it), A378606 (Möbius transform).
%Y A349387 Cf. also A000010, A000203, A003972, A003973, A347236, A349381.
%K A349387 nonn,mult
%O A349387 1,3
%A A349387 _Antti Karttunen_, Nov 17 2021