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 A349388 #19 Dec 11 2024 20:17:52
%S A349388 1,-1,-2,-2,-2,2,-4,-4,-6,2,-2,4,-4,4,4,-8,-2,6,-4,4,8,2,-6,8,-10,4,
%T A349388 -18,8,-2,-4,-6,-16,4,2,8,12,-4,4,8,8,-2,-8,-4,4,12,6,-6,16,-28,10,4,
%U A349388 8,-6,18,4,16,8,2,-2,-8,-6,6,24,-32,8,-4,-4,4,12,-8,-2,24,-6,4,20,8,8,-8,-4,16,-54,2,-6,-16,4,4,4
%N A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).
%C A349388 Multiplicative because A000027 and A346234 are.
%H A349388 Antti Karttunen, <a href="/A349388/b349388.txt">Table of n, a(n) for n = 1..20000</a>
%H A349388 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%H A349388 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349388 a(n) = Sum_{d|n} d * A346234(n/d).
%F A349388 For all n >= 1, a(A000040(n)) = -A001223(n).
%F A349388 Multiplicative with a(p^e) = p^e - nextprime(p) * p^(e-1), where nextprime function is A151800. - _Amiram Eldar_, Nov 18 2021
%t A349388 f[p_, e_] := p^e - NextPrime[p] * p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349388 (PARI)
%o A349388 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349388 A346234(n) = (moebius(n)*A003961(n));
%o A349388 A349388(n) = sumdiv(n,d,d*A346234(n/d));
%Y A349388 Cf. A000027, A000040, A001223, A003961, A151800, A346234, A349387 (Dirichlet inverse), A349389 (sum with it), A378607 (inverse Möbius transform).
%Y A349388 Cf. also A347238.
%K A349388 sign,mult
%O A349388 1,3
%A A349388 _Antti Karttunen_, Nov 17 2021