A379218 Möbius transform of A379108.
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, 29, 10, 30, 16, 22, 17, 30, 14, 37, 19, 26, 20, 41, 12, 43, 22, 35, 23, 47, 16, 43, 25, 34, 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
Offset: 1
Links
Crossrefs
Programs
-
Mathematica
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 *)
-
PARI
A209229(n) = (n && !bitand(n,n-1)); 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])); }; A379218(n) = sumdiv(n,d,d*A359579(n/d));
Formula
a(n) = Sum_{d|n} d*A359579(n/d).
From Amiram Eldar, Jan 02 2025: (Start)
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.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) / Product_{p in A000668} (1 + 1/p^2) = 0.33038569613198448017... . (End)
Comments