A340143 -id:A340143 - cp's OEIS frontend

A340142 Dirichlet inverse of A160595, where A160595(x) = phi(x)/gcd(phi(x), x-1).

1, -1, -1, -1, -1, 0, -1, -1, -2, -2, -1, 1, -1, -4, -2, -1, -1, 1, -1, 1, -1, -8, -1, 2, -4, -10, -4, 9, -1, 6, -1, -1, -3, -14, -10, 3, -1, -16, -10, 4, -1, 4, -1, 1, 5, -20, -1, 3, -6, -5, -14, 17, -1, 4, -18, -10, -7, -26, -1, 8, -1, -28, -9, -1, -1, 24, -1, 1, -9, 30, -1, 4, -1, -34, -5, 25, -13, 22, -1, 7, -8, -38, -1, -5

Offset: 1

Antti Karttunen, Dec 29 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. A160595.

Cf. also A340141, A340143.

up_to = 65537;
A160595(n) = { my(x=eulerphi(n)); x/gcd(x,n-1); };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA160595(n)));
A340142(n) = v340142[n];

A340141 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A160595(x) = phi(x)/gcd(phi(x), x-1).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 25, 11, 7, 1, 19, 1, 11, 7, 363, 1, 31, 1, 43, 5, 19, 1, 63, 19, 23, 61, 3, 1, 15, 1, 1335, 9, 31, 23, 189, 1, 35, 23, 139, 1, 29, 1, 115, 23, 43, 1, 867, 27, 127, 31, 11, 1, 163, 39, 279, 17, 55, 1, 73, 1, 59, 123, 9923, 5, -15, 1, 187, 21, -9, 1, 615, 1, 71, 127, 19, 29, 47, 1, 1875, 1363, 79, 1, 203, 31

Offset: 1

Antti Karttunen, Dec 29 2020

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

Cf. A046644 (denominators).
Cf. A160595.
Cf. also A340142, A340143.

up_to = 65537;
A160595(n) = { my(x=eulerphi(n)); x/gcd(x,n-1); };
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA160595(n)));
A340141(n) = numerator(v340141rat[n]);

A340146 Möbius transform of A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 2, 3, 0, 1, 0, 5, 1, 4, 0, 2, 0, 3, 2, 9, 0, 2, 4, 11, 6, -3, 0, 2, 0, 8, 4, 15, 5, 4, 0, 17, 5, 6, 0, 3, 0, 9, -1, 21, 0, 4, 6, 12, 7, -5, 0, 6, 9, 18, 8, 27, 0, 3, 0, 29, 4, 16, 2, -11, 0, 15, 10, -6, 0, 8, 0, 35, 4, -7, 14, 6, 0, 12, 18, 39, 0, 13, 3, 41, 13, 18, 0, 13, 1, 21, 14, 45, 17, 8

Offset: 1

Antti Karttunen, Dec 29 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, A247074.

Cf. also A340143, A340144, A340145.

A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
A340146(n) = sumdiv(n,d,moebius(n/d)*A247074(d));

a(n) = Sum_{d|n} A008683(n/d) * A247074(d).

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

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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).

cp's OEIS Frontend

A340142 Dirichlet inverse of A160595, where A160595(x) = phi(x)/gcd(phi(x), x-1).

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A340141 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A160595(x) = phi(x)/gcd(phi(x), x-1).

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A340146 Möbius transform of A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

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

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