A318828 -id:A318828 - cp's OEIS frontend

A318829 a(n) = A063994(n) / A049559(n) = (1/gcd(n-1, phi(n))) * Product_{primes p dividing n} gcd(p-1, n-1).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2

Offset: 1

Antti Karttunen, Sep 09 2018

nonn

Records occur at: 1, 15, 85, 247, 671, 949, 1105, 1387, 2047, 2821, 9471, 11305, 13747, 13981, 29341, 40885, 51319, 63973, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A049559, A063994, A160595, A247074, A318828.

A049559(n) = gcd(eulerphi(n), n-1); \\ From A049559
A063994(n) = { my(f=factor(n)[,1]); prod(i=1, #f, gcd(f[i]-1, n-1)); };
A318829(n) = (A063994(n)/A049559(n));

a(n) = A063994(n) / A049559(n).

a(n) = A160595(n) / A247074(n).

A318827 a(n) = n - gcd(n - 1, phi(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 7, 9, 1, 11, 1, 13, 13, 15, 1, 17, 1, 19, 17, 21, 1, 23, 21, 25, 25, 25, 1, 29, 1, 31, 29, 33, 33, 35, 1, 37, 37, 39, 1, 41, 1, 43, 41, 45, 1, 47, 43, 49, 49, 49, 1, 53, 53, 55, 53, 57, 1, 59, 1, 61, 61, 63, 49, 61, 1, 67, 65, 67, 1, 71, 1, 73, 73, 73, 73, 77, 1, 79, 79, 81, 1, 83, 81, 85, 85, 87, 1, 89, 73

Offset: 1

Antti Karttunen, Sep 09 2018

nonn

a(n) = n-1 for n in A209211.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A051953, A049559, A209211, A318828, A318829, A318830.

Array[# - GCD[# - 1, EulerPhi[#]] &, 100] (* Alonso del Arte, Sep 09 2018 *)

PARI
```
A318827(n) = (n-gcd(eulerphi(n), n-1));
```

a(n) = n - A049559(n) = n - gcd(n - 1, phi(n)).
a(n) = A051953(n) + A318830(n).
a(p) = 1 for p prime.

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).

A340189 a(n) = n + A340187(n).

Original entry on oeis.org

2, 1, 1, 4, 1, 9, 1, 8, 11, 17, 1, 11, 1, 25, 27, 16, 1, 13, 1, 17, 41, 41, 1, 24, 37, 49, 25, 21, 1, 1, 1, 32, 69, 65, 79, 40, 1, 73, 83, 40, 1, -7, 1, 35, 21, 89, 1, 48, 79, 17, 111, 39, 1, 61, 131, 60, 125, 113, 1, 83, 1, 121, 27, 64, 145, -27, 1, 53, 153, -49, 1, 71, 1, 145, 23, 57, 193, -31, 1, 80, 83, 161, 1, 131

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, A340188.

Cf. also A318833.

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];
A340189(n) = (n+A340187(n));

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

a(n) = A340188(n) + A318828(n).

A318840 a(n) = phi(n) - Product_{primes p dividing n} gcd(p-1, n-1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 4, 3, 0, 3, 0, 5, 4, 7, 0, 5, 0, 7, 8, 9, 0, 7, 16, 11, 16, 9, 0, 7, 0, 15, 16, 15, 20, 11, 0, 17, 20, 15, 0, 11, 0, 19, 16, 21, 0, 15, 36, 19, 28, 21, 0, 17, 36, 23, 32, 27, 0, 15, 0, 29, 32, 31, 32, 15, 0, 31, 40, 21, 0, 23, 0, 35, 36, 33, 56, 23, 0, 31, 52, 39, 0, 23, 48, 41, 52, 39, 0, 23, 36, 43, 56, 45, 68, 31

Offset: 1

Antti Karttunen, Sep 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A063994, A318828, A318830.

A063994(n) = { my(f=factor(n)[,1]); prod(i=1, #f, gcd(f[i]-1, n-1)); };
A318840(n) = (eulerphi(n) - A063994(n));

a(n) = A000010(n) - A063994(n).

a(n) = A318828(n) - A051953(n).

cp's OEIS Frontend

A318829 a(n) = A063994(n) / A049559(n) = (1/gcd(n-1, phi(n))) * Product_{primes p dividing n} gcd(p-1, n-1).

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A318827 a(n) = n - gcd(n - 1, phi(n)).

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

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A340189 a(n) = n + A340187(n).

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A318840 a(n) = phi(n) - Product_{primes p dividing n} gcd(p-1, n-1).

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