A349616 -id:A349616 - cp's OEIS frontend

A349613 Dirichlet convolution of A064413 (EKG-permutation) with the Dirichlet inverse of its inverse permutation.

1, 0, -1, 3, -7, 7, -2, -6, 9, 10, -5, -15, -14, -2, 55, 10, -17, -41, -15, -36, 42, 18, -13, 44, 81, 29, -35, -45, -18, -180, -29, -23, 41, 53, 135, 99, -48, 51, 114, 131, -30, -140, -58, -53, -303, 34, -37, -120, 34, -196, 147, -87, -45, 226, 207, 166, 103, 67, -41, 466, -84, 91, -288, 13, 350, -258, -91, -108

Offset: 1

Antti Karttunen, Nov 23 2021

sign

Obviously, convolving this with A064664 gives A064413 back.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A064413, A064664, A323411, A349614 (Dirichlet inverse), A349615 (sum with it), A349616.

Cf. also pairs A349376, A349377 and A349397, A349398 for similar constructions.

up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA064664(n)));
A323411(n) = v323411[n];
A349613(n) = sumdiv(n,d,A064413(d)*A323411(n/d));

a(n) = Sum_{d|n} A064413(d) * A323411(n/d).

A349617 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 2, -1, 5, -6, 7, 2, -9, -11, 9, 2, 15, -15, -29, 1, 16, 18, 18, 5, -41, -21, 20, -4, -26, -29, 4, 7, 28, 64, 30, -3, -61, -34, -80, 9, 30, -38, -81, -6, 33, 92, 38, 14, 51, -44, 42, 10, -48, 53, -99, 6, 47, 4, -102, -17, -111, -58, 48, -4, 54, -62, 69, 2, -151, 146, 61, 18, -131, 157, 63, -3, 65, -68, 92, 18

Offset: 1

Antti Karttunen, Nov 23 2021

Dirichlet convolution of this sequence with A000010 (Euler phi) is A304526 (Möbius transform of the inverse permutation of EKG-sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A055615, A064413, A064664, A349616 (Dirichlet inverse).

Cf. also A000010, A304526, A349614.

A055615(n) = (n*moebius(n));
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
A349617(n) = sumdiv(n,d,A064664(d)*A055615(n/d));

a(n) = Sum_{d|n} A064664(d) * A055615(n/d).

cp's OEIS Frontend

A349613 Dirichlet convolution of A064413 (EKG-permutation) with the Dirichlet inverse of its inverse permutation.

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