A340142 -id:A340142 - cp's OEIS frontend

A340143 Möbius transform of A160595, where A160595(x) = phi(x)/gcd(phi(x), x-1).

1, 0, 0, 1, 0, 1, 0, 2, 2, 3, 0, 1, 0, 5, 3, 4, 0, 2, 0, 3, 2, 9, 0, 2, 4, 11, 6, -3, 0, 0, 0, 8, 4, 15, 11, 4, 0, 17, 11, 6, 0, 3, 0, 9, 0, 21, 0, 4, 6, 12, 15, -5, 0, 6, 19, 18, 8, 27, 0, 3, 0, 29, 13, 16, 2, -11, 0, 15, 10, -12, 0, 8, 0, 35, 12, -7, 14, 0, 0, 12, 18, 39, 0, 13, 15, 41, 27, 18, 0, 12, 3, 21, 14, 45, 35

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. A008683, A160595.

Cf. also A340141, A340142.

Table[DivisorSum[n, MoebiusMu[n/#1]*#2/GCD[#2, #3] & @@ {#, EulerPhi[#], # - 1} &], {n, 95}] (* Michael De Vlieger, Dec 29 2020 *)

A160595(n) = { my(x=eulerphi(n)); x/gcd(x,n-1); };
A340143(n) = sumdiv(n,d,moebius(n/d)*A160595(d));

a(n) = Sum_{d|n} A008683(n/d) * A160595(d).

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]);

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

Original entry on oeis.org

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];

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

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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];

cp's OEIS Frontend

A340143 Möbius transform 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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A340145 Dirichlet inverse of A247074(x) = phi(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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