A340146 -id:A340146 - cp's OEIS frontend

A340145 Dirichlet inverse of A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

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

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. A247074.

Cf. also A340142, A340144, A340146.

up_to = 65537;
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA247074(n)));
A340145(n) = v340145[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).

A340144 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 25, 11, 7, 1, 19, 1, 11, 3, 363, 1, 31, 1, 43, 5, 19, 1, 63, 19, 23, 61, 3, 1, 19, 1, 1335, 9, 31, 11, 189, 1, 35, 11, 139, 1, 29, 1, 115, 7, 43, 1, 867, 27, 127, 15, 11, 1, 163, 19, 279, 17, 55, 1, 93, 1, 59, 51, 9923, 5, -15, 1, 187, 21, 3, 1, 615, 1, 71, 55, 19, 29, 59, 1, 1875, 1363, 79, 1, 203

Offset: 1

Antti Karttunen, Dec 29 2020

			For n = 561 = 3*11*17, its divisors d are: 1, 3, 11, 17, 33, 51, 187, 561.
For this sequence, the corresponding terms a(d) are: 1, 1, 1, 1, 9, 15, 79, -99.
For A046644, the corresponding terms are:            1, 2, 2, 2, 4,  4,  4,   8.
Convolving these ratios as Sum_{d|561} r(d)*r(n/d) = 2*((1/1)*(-99/8) + (1/2)*(79/4) + (1/2)*(15/4) + (1/2)*(9/4)) yields 1 as expected, because 561 is Carmichael number (A002997) and A247074 obtains value 1 on all of them.

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. A247074.
Cf. also A340141, A340145, A340146.

up_to = 65537;
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
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&&dA247074(n)));
A340144(n) = numerator(v340144rat[n]);

cp's OEIS Frontend

A340145 Dirichlet inverse of A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A340144 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI