A347231 -id:A347231 - cp's OEIS frontend

A347229 Sum of A344695 [gcd(sigma(n), psi(n))] and its Dirichlet inverse.

2, 0, 0, 9, 0, 24, 0, -21, 16, 36, 0, -28, 0, 48, 48, 73, 0, -42, 0, -42, 64, 72, 0, 108, 36, 84, -56, -56, 0, 0, 0, -213, 96, 108, 96, 121, 0, 120, 112, 162, 0, 0, 0, -84, -84, 144, 0, -284, 64, -102, 144, -98, 0, 192, 144, 216, 160, 180, 0, 216, 0, 192, -112, 649, 168, 0, 0, -126, 192, 0, 0, -357, 0, 228, -136

Offset: 1

Antti Karttunen, Aug 25 2021

sign

It seems that A030059 gives the positions of all zeros.

Antti Karttunen, Table of n, a(n) for n = 1..12000

Cf. A000203, A001615, A030059, A030229, A344695, A347228, A347231.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA001615(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));
v347228 = DirInverseCorrect(vector(up_to,n,A344695(n)));
A347228(n) = v347228[n];
A347229(n) = (A344695(n)+A347228(n));

a(n) = A344695(n) + A347228(n).
a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A344695(d) * A347228(n/d).
For all n >= 1, a(A030059(n)) = 0 and a(A030229(n)) = 2*A344695(A030229(n)). [Even though A344695 is not multiplicative, this holds because on squarefree n it is equal to psi(n) and sigma(n) that are multiplicative functions]

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

Original entry on oeis.org

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

Antti Karttunen, Aug 25 2021

sign

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.

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

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

Cf. also A007947, A063659.

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));

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

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

cp's OEIS Frontend

A347229 Sum of A344695 [gcd(sigma(n), psi(n))] and its Dirichlet inverse.

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