A379108 -id:A379108 - cp's OEIS frontend

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

Antti Karttunen, Dec 18 2024

Dirichlet convolution of A000027 with A359579.

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences related to sigma(n).

Cf. A000027, A000203, A000668, A008683, A336923, A359579, A379108, A379219 (Dirichlet inverse).

Cf. also A026741.

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 *)

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));

a(n) = Sum_{d|n} d*A359579(n/d).
a(n) = Sum_{d|n} A008683(d)*A379108(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)

A379109 Dirichlet convolution of A046692 (inverse of sigma) with A336923, where A336923(n) = 1 if sigma(2n) - sigma(n) is a power of 2, otherwise 0.

Original entry on oeis.org

1, -2, -3, 0, -6, 6, -7, 0, -1, 12, -12, 0, -14, 14, 18, 0, -18, 2, -20, 0, 21, 24, -24, 0, 5, 28, 3, 0, -30, -36, -31, 0, 36, 36, 42, 0, -38, 40, 42, 0, -42, -42, -44, 0, 6, 48, -48, 0, -1, -10, 54, 0, -54, -6, 72, 0, 60, 60, -60, 0, -62, 62, 7, 0, 84, -72, -68, 0, 72, -84, -72, 0, -74, 76, -15, 0, 84, -84, -80, 0, 0, 84

Offset: 1

Antti Karttunen, Dec 17 2024

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences related to sigma(n).

Cf. A000203, A000668, A046692, A054784, A336923, A379108 (Dirichlet inverse).

f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, Which[e == 1, -p, e == 2, -1, e == 3, p, e > 3, 0], Which[e == 1, -p - 1, e == 2, p, e > 2, 0]]; f[2, e_] := If[e == 1, -2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 03 2025 *)

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))); };
A209229(n) = (n && !bitand(n,n-1));
A336923(n) = A209229(sigma(n+n)-sigma(n));
A379109(n) = sumdiv(n,d,A046692(d)*A336923(n/d));

a(n) = Sum_{d|n} A046692(d)*A336923(n/d).

Multiplicative with a(2^e) = -2 if e = 1 and 0 otherwise, and for an odd prime p, if p is a Mersenne prime, a(p) = -p, a(p^2) = -1, a(p^3) = p, and a(p^e) = 0 for e >= 4, and otherwise a(p) = -(p+1), a(p^2) = p and a(p^e) = 0 for e >= 3. - Amiram Eldar, Jan 03 2025

cp's OEIS Frontend

A379218 Möbius transform of A379108.

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