A340141 -id:A340141 - 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];

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

Original entry on oeis.org

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

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, 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).

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]);

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

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

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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)).

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