A340192 -id:A340192 - 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

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, 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).

A340193 a(n) = n - (Sum_{d|n} A063994(d)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 4, 4, 3, 0, 5, 0, 5, 4, 11, 0, 10, 0, 11, 8, 9, 0, 15, 16, 11, 20, 15, 0, 15, 0, 26, 16, 15, 20, 25, 0, 17, 20, 29, 0, 25, 0, 29, 24, 21, 0, 37, 36, 38, 28, 33, 0, 43, 36, 41, 32, 27, 0, 41, 0, 29, 44, 57, 32, 41, 0, 47, 40, 49, 0, 58, 0, 35, 56, 51, 56, 55, 0, 67, 72, 39, 0, 61, 48, 41, 52, 71, 0, 63, 36

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 A318840.
Cf. A063994, A340192.
Cf. also A051953.

A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
A340193(n) = (n-sumdiv(n,d,A063994(d)));

a(n) = n - A340192(n).

a(n) = Sum_{d|n} A318840(d).

A340195 a(n) = Sum_{divisors d of n} A049559(d), where A049559(x) = gcd(phi(x), x-1).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 7, 4, 5, 7, 11, 7, 13, 9, 9, 5, 17, 8, 19, 9, 13, 13, 23, 9, 9, 15, 7, 13, 29, 13, 31, 6, 17, 19, 13, 11, 37, 21, 17, 11, 41, 17, 43, 15, 15, 25, 47, 11, 13, 12, 21, 19, 53, 11, 17, 15, 25, 31, 59, 17, 61, 33, 17, 7, 33, 25, 67, 21, 29, 19, 71, 14, 73, 39, 15, 25, 21, 21, 79, 13, 9, 43, 83, 23, 25, 45, 33

Offset: 1

Antti Karttunen, Jan 02 2021

nonn

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

Inverse Möbius transform of A049559.
Cf. A000010, A340196.
Cf. also A340192.

A049559(n) = gcd(eulerphi(n), n-1);
A340195(n) = sumdiv(n,d,A049559(d));

a(n) = Sum_{d|n} A049559(d).

a(n) = n - A340196(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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A340193 a(n) = n - (Sum_{d|n} A063994(d)).

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Formula

A340195 a(n) = Sum_{divisors d of n} A049559(d), where A049559(x) = gcd(phi(x), x-1).

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