A349351 -id:A349351 - cp's OEIS frontend

A349377 Dirichlet convolution of A006369 with the Dirichlet inverse of A006368, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

1, 0, 0, -1, 3, -5, 4, 2, -1, -11, 7, 7, 7, -14, -7, -3, 10, -2, 11, 16, -10, -25, 14, -6, 2, -25, 0, 18, 17, 11, 18, 4, -17, -36, -10, 20, 21, -39, -17, -18, 24, 12, 25, 34, -7, -50, 28, 2, 8, -15, -24, 34, 31, 3, -20, -16, -27, -61, 35, 30, 35, -64, -8, -5, -20, 23, 39, 50, -34, 6, 42, -44, 42, -75, -15, 52, -22, 23

Offset: 1

Antti Karttunen, Nov 17 2021

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Obviously, convolving this sequence with A006368 gives its inverse A006369 from n >= 1 onward.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006368, A006369, A349351, A349376 (Dirichlet inverse), A349378 (sum with it).

Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349351 = Map();
A349351(n) = if(1==n,1,my(v); if(mapisdefined(memoA349351,n,&v), v, v = -sumdiv(n,d,if(dA006368(n/d)*A349351(d),0)); mapput(memoA349351,n,v); (v)));
A349377(n) = sumdiv(n,d,A006369(d)*A349351(n/d));

a(n) = Sum_{d|n} A006369(d) * A349351(n/d).

a(n) = A349378(n) - A349376(n).

A349352 Sum of A006368, "the amusical permutation", and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 12, 0, 9, 4, 24, 0, 24, 0, 30, 16, 27, 0, 42, 0, 30, 20, 48, 0, 21, 16, 60, 20, 51, 0, 54, 0, 45, 32, 78, 40, 30, 0, 84, 40, 51, 0, 90, 0, 78, 52, 102, 0, 96, 25, 90, 52, 84, 0, 90, 64, 57, 56, 132, 0, 27, 0, 138, 74, 99, 80, 138, 0, 111, 68, 114, 0, 114, 0, 168, 68, 132, 80, 150, 0, 138, 61, 186, 0

Offset: 1

Antti Karttunen, Nov 15 2021

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The first negative term is a(2520) = -918.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006368, A349351, A349369, A349378.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
v349351 = DirInverseCorrect(vector(up_to,n,A006368(n)));
A349351(n) = v349351[n];
A349352(n) = (A006368(n)+A349351(n));

a(n) = A006368(n) + A349351(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A006368(d) * A349351(n/d).

A349368 Dirichlet inverse of A006369, the inverse permutation of "amusical permutation".

Original entry on oeis.org

1, -3, -2, 4, -7, 8, -9, -8, -2, 29, -15, -18, -17, 35, 18, 16, -23, 4, -25, -68, 22, 61, -31, 44, 16, 67, -2, -76, -39, -104, -41, -32, 38, 93, 79, 6, -49, 99, 42, 168, -55, -120, -57, -140, 10, 125, -63, -104, 16, -128, 58, -148, -71, 0, 137, 184, 62, 157, -79, 354, -81, 163, 14, 64, 151, -216, -89, -212, 78, -445

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006369, A349369.

Cf. also A349351, A349376.

A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349368 = Map();
A349368(n) = if(1==n,1,my(v); if(mapisdefined(memoA349368,n,&v), v, v = -sumdiv(n,d,if(dA006369(n/d)*A349368(d),0)); mapput(memoA349368,n,v); (v)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A006369(n/d) * a(d).

a(n) = A349369(n) - A006369(n).

cp's OEIS Frontend

A349377 Dirichlet convolution of A006369 with the Dirichlet inverse of A006368, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

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A349352 Sum of A006368, "the amusical permutation", and its Dirichlet inverse.

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A349368 Dirichlet inverse of A006369, the inverse permutation of "amusical permutation".

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