A063994 -id:A063994 - cp's OEIS frontend

A318828 a(n) = n - A063994(n) = n - Product_{primes p dividing n} gcd(p-1, n-1).

0, 1, 1, 3, 1, 5, 1, 7, 7, 9, 1, 11, 1, 13, 11, 15, 1, 17, 1, 19, 17, 21, 1, 23, 21, 25, 25, 25, 1, 29, 1, 31, 29, 33, 31, 35, 1, 37, 35, 39, 1, 41, 1, 43, 37, 45, 1, 47, 43, 49, 47, 49, 1, 53, 51, 55, 53, 57, 1, 59, 1, 61, 59, 63, 49, 61, 1, 67, 65, 67, 1, 71, 1, 73, 71, 73, 73, 77, 1, 79, 79, 81, 1, 83, 69, 85, 83, 87, 1, 89, 55

Offset: 1

Antti Karttunen, Sep 09 2018

nonn

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

Cf. A063994, A318827, A318829.

A063994(n) = { my(f=factor(n)[,1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
A318828(n) = (n-A063994(n));

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

A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 15, 16, 3, 17, 18, 19, 20, 21, 3, 22, 3, 23, 24, 25, 26, 27, 3, 28, 26, 29, 3, 30, 3, 31, 32, 33, 3, 34, 35, 36, 37, 38, 3, 39, 40, 41, 42, 43, 3, 44, 3, 45, 46, 47, 48, 49, 3, 50, 51, 52, 3, 53, 3, 54, 55, 56, 57, 58, 3, 59, 60, 61, 3, 62, 63, 64, 65, 66, 3, 67, 68, 69, 57, 70, 71, 72, 3, 73, 74, 75, 3

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323371(i) = A323371(j),
  a(i) = a(j) => A247074(i) = A247074(j).

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

Cf. A000010, A003557, A023900, A063994, A247074, A305801, A323371, A323404.
Differs from A323370 for the first time at n=78, where a(78) = 58, while A323370(78) = 52.
Cf. also A323374.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
Aux323405(n) = if((n>2)&&isprime(n),0,[A003557(n), A023900(n), A063994(n)]);
v323405 = rgs_transform(vector(up_to, n, Aux323405(n)));
A323405(n) = v323405[n];

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

Original entry on oeis.org

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).

A323404 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 35, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 76, 92

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, j: a(i) = a(j) => A247074(i) = A247074(j).

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

Cf. A000010, A003557, A023900, A063994, A247074, A323405.

Cf. also A323373.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
Aux323404(n) = if(1,[A003557(n), A023900(n), A063994(n)]);
v323404 = rgs_transform(vector(up_to, n, Aux323404(n)));
A323404(n) = v323404[n];

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

Original entry on oeis.org

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).

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

Original entry on oeis.org

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];

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).

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).

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).

cp's OEIS Frontend

A318828 a(n) = n - A063994(n) = n - Product_{primes p dividing n} gcd(p-1, n-1).

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A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

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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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PARI

Formula

A323404 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)].

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PARI

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

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PARI

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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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PARI

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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A340191 Difference between A063994 and its Möbius transform.

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