A340190 -id:A340190 - cp's OEIS frontend

A340192 a(n) = Sum_{d|n} A063994(d), where A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

1, 2, 3, 3, 5, 5, 7, 4, 5, 7, 11, 7, 13, 9, 11, 5, 17, 8, 19, 9, 13, 13, 23, 9, 9, 15, 7, 13, 29, 15, 31, 6, 17, 19, 15, 11, 37, 21, 19, 11, 41, 17, 43, 15, 21, 25, 47, 11, 13, 12, 23, 19, 53, 11, 19, 15, 25, 31, 59, 19, 61, 33, 19, 7, 33, 25, 67, 21, 29, 21, 71, 14, 73, 39, 19, 25, 21, 23, 79, 13, 9, 43, 83, 23, 37

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

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

Inverse Möbius transform of A063994.

Cf. A340190, A340193.

A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
A340192(n) = sumdiv(n,d,A063994(d));

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

a(n) = n - A340193(n).

A340187 Dirichlet inverse of A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

Original entry on oeis.org

1, -1, -2, 0, -4, 3, -6, 0, 2, 7, -10, -1, -12, 11, 12, 0, -16, -5, -18, -3, 20, 19, -22, 0, 12, 23, -2, -7, -28, -29, -30, 0, 36, 31, 44, 4, -36, 35, 44, 0, -40, -49, -42, -9, -24, 43, -46, 0, 30, -33, 60, -13, -52, 7, 76, 4, 68, 55, -58, 23, -60, 59, -36, 0, 80, -93, -66, -15, 84, -119, -70, -1, -72, 71, -52, -19

Offset: 1

Antti Karttunen, Dec 31 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A063994, A340188, A340189.

Cf. also A023900, A340142, A340145, A340190.

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];

A340191 Difference between A063994 and its Möbius transform.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 6, 5, 1, 1, 1, 1, 1, 7, 10, 1, 1, 4, 12, 2, 1, 1, 0, 1, 1, 11, 16, 9, 1, 1, 18, 13, 1, 1, -2, 1, 1, 4, 22, 1, 1, 6, 1, 17, 1, 1, 1, 13, 3, 19, 28, 1, 1, 1, 30, 4, 1, 15, -6, 1, 1, 23, -4, 1, 1, 1, 36, 4, 1, 15, -8, 1, 1, 2, 40, 1, 3, 19, 42, 29, 1, 1, 5, 17, 1, 31, 46, 21

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. A008683, A063994, A340190.

A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
A340191(n) = -sumdiv(n,d,(dA063994(d));

a(n) = A063994(n) - A340190(n).

a(n) = -Sum_{d|n, dA008683(n/d) * A063994(d).

cp's OEIS Frontend

A340192 a(n) = Sum_{d|n} A063994(d), where A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

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A340187 Dirichlet inverse of A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

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A340191 Difference between A063994 and its Möbius transform.

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