A340191 -id:A340191 - cp's OEIS frontend

A340190 Möbius transform of A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

1, 0, 1, 0, 3, -1, 5, 0, 0, -3, 9, 0, 11, -5, -1, 0, 15, 0, 17, 0, -3, -9, 21, 0, 0, -11, 0, 2, 27, 1, 29, 0, -7, -15, -5, 0, 35, -17, -9, 0, 39, 3, 41, 0, 4, -21, 45, 0, 0, 0, -13, 2, 51, 0, -9, -2, -15, -27, 57, 0, 59, -29, 0, 0, 1, 11, 65, 0, -19, 7, 69, 0, 71, -35, 0, 2, -11, 9, 77, 0, 0, -39, 81, -2, -3, -41, -25

Offset: 1

Antti Karttunen, Dec 31 2020

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Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008683, A063994, A340191.

Cf. also A007431, A340143, A340146, A340192.

A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
A340190(n) = sumdiv(n,d,moebius(n/d)*A063994(d));

a(n) = Sum_{d|n} A008683(n/d) * A063994(d).

a(n) = A063994(n) - A340191(n).

A340188 Sum of A063994 and its Dirichlet inverse, where A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 8, 0, 0, 0, 12, 16, 1, 0, -4, 0, -2, 24, 20, 0, 1, 16, 24, 0, -4, 0, -28, 0, 1, 40, 32, 48, 5, 0, 36, 48, 1, 0, -48, 0, -8, -16, 44, 0, 1, 36, -32, 64, -10, 0, 8, 80, 5, 72, 56, 0, 24, 0, 60, -32, 1, 96, -88, 0, -14, 88, -116, 0, 0, 0, 72, -48, -16, 120, -108, 0, 1, 4, 80, 0, 48, 128, 84, 112

Offset: 1

Antti Karttunen, Dec 31 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A063994, A318828, A340187, A340189.

Cf. also A319340, A340191.

up_to = 65537;
A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA063994(n)));
A340187(n) = v340187[n];
A340188(n) = (A063994(n)+A340187(n));

a(n) = A063994(n) + A340187(n).

a(n) = A340189(n) - A318828(n).

cp's OEIS Frontend

A340190 Möbius transform of A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

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