A347230 -id:A347230 - cp's OEIS frontend

A347228 Dirichlet inverse of A344695, gcd(sigma(n), psi(n)).

1, -3, -4, 8, -6, 12, -8, -24, 15, 18, -12, -32, -14, 24, 24, 72, -18, -45, -20, -48, 32, 36, -24, 96, 35, 42, -60, -64, -30, -72, -32, -216, 48, 54, 48, 120, -38, 60, 56, 144, -42, -96, -44, -96, -90, 72, -48, -288, 63, -105, 72, -112, -54, 180, 72, 192, 80, 90, -60, 192, -62, 96, -120, 648, 84, -144, -68, -144, 96

Offset: 1

Antti Karttunen, Aug 25 2021

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This is not multiplicative because A344695 is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = -484 <> -480 = 8 * -60 = a(4) * a(27).

Conjecture: No zeros occur as terms. Checked up to n = 2^21.

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

Cf. A000203, A001615, A344695, A347229, A347230.

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];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A344695(n/d) * a(d).

a(n) = A347229(n) - A344695(n).

A347231 Difference between A344695 [gcd(psi(n),sigma(n))] and its Möbius transform.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 1, 4, 8, 1, 10, 1, 10, 9, 3, 1, 9, 1, 16, 11, 14, 1, 6, 6, 16, 1, 22, 1, 42, 1, 1, 15, 20, 13, -5, 1, 22, 17, 8, 1, 54, 1, 34, 21, 26, 1, 10, 8, 13, 21, 40, 1, 6, 17, 10, 23, 32, 1, 54, 1, 34, 29, 3, 19, 78, 1, 52, 27, 74, 1, 9, 1, 40, 19, 58, 19, 90, 1, 16, 4, 44, 1, 74, 23, 46, 33, 14, 1, 48

Offset: 1

Antti Karttunen, Aug 25 2021

sign

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

Cf. A000203, A001615, A008683, A344695, A347230.

Cf. also A347229.

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));
A347231(n) = -sumdiv(n,d,(dA344695(d));

a(n) = A344695(n) - A347230(n).

a(n) = -Sum_{d|n, dA008683(n/d) * A344695(d).

cp's OEIS Frontend

A347228 Dirichlet inverse of A344695, gcd(sigma(n), psi(n)).

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