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 A379218 #14 Jan 03 2025 05:16:05
%S A379218 1,1,2,2,5,2,6,4,7,5,11,4,13,6,10,8,17,7,19,10,12,11,23,8,25,13,20,12,
%T A379218 29,10,30,16,22,17,30,14,37,19,26,20,41,12,43,22,35,23,47,16,43,25,34,
%U A379218 26,53,20,55,24,38,29,59,20,61,30,42,32,65,22,67,34,46,30,71,28,73,37,50,38,66,26,79,40,61,41,83,24
%N A379218 Möbius transform of A379108.
%C A379218 Dirichlet convolution of A000027 with A359579.
%H A379218 Antti Karttunen, <a href="/A379218/b379218.txt">Table of n, a(n) for n = 1..32769</a>
%H A379218 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379218 a(n) = Sum_{d|n} d*A359579(n/d).
%F A379218 a(n) = Sum_{d|n} A008683(d)*A379108(n/d).
%F A379218 From _Amiram Eldar_, Jan 02 2025: (Start)
%F A379218 Multiplicative with a(2^e) = 2^(e-1), and for an odd prime p, a(p^e) = (p^(e + 1) + (-1)^e)/(p + 1) if p is a Mersenne prime (A000668), and a(p^e) = p^e otherwise.
%F A379218 Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) / Product_{p in A000668} (1 + 1/p^2) = 0.33038569613198448017... . (End)
%t A379218 f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, (p^(e + 1) + (-1)^e)/(p + 1), p^e]; f[2, e_] := 2^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 02 2025 *)
%o A379218 (PARI)
%o A379218 A209229(n) = (n && !bitand(n,n-1));
%o A379218 A359579(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1], -(1==f[k,2]), (-A209229(1+f[k,1]))^f[k,2])); };
%o A379218 A379218(n) = sumdiv(n,d,d*A359579(n/d));
%Y A379218 Cf. A000027, A000203, A000668, A008683, A336923, A359579, A379108, A379219 (Dirichlet inverse).
%Y A379218 Cf. also A026741.
%K A379218 nonn,mult
%O A379218 1,3
%A A379218 _Antti Karttunen_, Dec 18 2024