A347230 Möbius transform of A344695, gcd(sigma(n), psi(n)).

1, 2, 3, -2, 5, 6, 7, 2, -3, 10, 11, -6, 13, 14, 15, -2, 17, -6, 19, -10, 21, 22, 23, 6, -5, 26, 3, -14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, -22, -15, 46, 47, -6, -7, -10, 51, -26, 53, 6, 55, 14, 57, 58, 59, -30, 61, 62, -21, -2, 65, 66, 67, -34, 69, 70, 71, -6, 73, 74, -15, -38, 77, 78, 79

Offset: 1

Author

Antti Karttunen, Aug 25 2021

Keywords

• sign

Comments

Not multiplicative because A344695 isn't either. For example, a(4) = -2, a(27) = 3, but a(108) = -2 != -6.

The absolute values are not equal to A007947. The first n where abs(a(n)) != A007947(n) is at n=108, with a(108) = -2, while A007947(108) = 6.

The first n such that a(n) does not divide n are: 196, 216, 392, 432, 441, 588, etc.

The zeros occur at n = 288, 576, 1440, 2016, 2880, 3168, 3744, 4032, etc.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000203, A001615, A008683, A344695, A347228, A347231.

Cf. also A007947, A063659.

Programs

PARI

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
A347230(n) = sumdiv(n,d,moebius(n/d)*A344695(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A344695(d).

a(n) = A344695(n) - A347231(n).