cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A323412 Sum of the inverse permutation of EKG-sequence, A064664, and its Dirichlet inverse, A323411.

Original entry on oeis.org

2, 0, 0, 4, 0, 20, 0, 4, 25, 40, 0, -14, 0, 56, 100, 21, 0, -86, 0, -24, 140, 80, 0, 64, 100, 112, -65, -32, 0, -386, 0, 30, 200, 132, 280, 233, 0, 148, 280, 138, 0, -538, 0, -44, -520, 172, 0, -55, 196, -324, 330, -60, 0, 596, 400, 194, 370, 228, 0, 898, 0, 244, -732, 67, 560, -766, 0, -70, 430, -1068, 0, -380, 0, 268, -1040, -78, 560
Offset: 1

Views

Author

Antti Karttunen, Jan 13 2019

Keywords

Links

Crossrefs

Cf. A064664, A323411.
Cf. also A304527, A323365.

Programs

A304526 Möbius transform of A064664, the inverse of EKG-sequence.

Original entry on oeis.org

1, 1, 4, 1, 9, -2, 13, 5, 1, -2, 19, 2, 27, -2, -3, 9, 32, 7, 36, 8, -3, -2, 42, 4, 14, -2, 16, 12, 56, 15, 60, 14, -3, -2, 2, 15, 66, -2, -3, 17, 73, 23, 80, 21, 27, -2, 88, 20, 36, 23, -3, 19, 99, 20, 24, 18, -3, -2, 106, 18, 114, -2, 35, 33, 17, 41, 127, 32, -3, 33, 133, 24, 137, -2, 44, 34, 45, 44, 150, 23, 54, -2, 159, 27, 42, -2, -3, 33, 166, 25, 52
Offset: 1

Views

Author

Antti Karttunen, May 18 2018

Keywords

Links

Crossrefs

Cf. A000010, A008683, A064423, A064664, A304527, A327867 (even bisection), A323411, A349617.

Programs

Formula

a(n) = Sum_{d|n} A008683(n/d)*A064664(d).
a(n) = A064664(n) - A304527(n).
For all n >= 1, a(A000040(n)) = A064423(n).
For n >= 2, a(2*A000040(n)) = -2.
For n >= 3, a(3*A000040(n)) = -3.
a(n) = Sum_{d|n} A000010(n/d)*A349617(d). - Antti Karttunen, Jan 27 2024

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Comments

Obviously, convolving this with A064664 gives A064413 back.

Links

Crossrefs

Cf. A064413, A064664, A323411, A349614 (Dirichlet inverse), A349615 (sum with it), A349616.
Cf. also pairs A349376, A349377 and A349397, A349398 for similar constructions.

Programs

  • PARI
    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));

Formula

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

A349616 Dirichlet convolution of A000027 (the identity function) with the Dirichlet inverse of the inverse permutation of EKG-permutation.

Original entry on oeis.org

1, 0, -2, 1, -5, 6, -7, -2, 13, 11, -9, -6, -15, 15, 49, 0, -16, -42, -18, -15, 69, 21, -20, 24, 51, 29, -48, -21, -28, -168, -30, -1, 97, 34, 150, 65, -30, 38, 141, 48, -33, -236, -38, -32, -317, 44, -42, -40, 97, -163, 163, -36, -47, 248, 192, 75, 183, 58, -48, 294, -54, 62, -443, 1, 301, -338, -61, -50, 211
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A000027, A064413, A064664, A323411, A349617 (Dirichlet inverse).
Cf. also A349613, A349614.

Programs

  • PARI
    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 was 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];
A349616(n) = sumdiv(n,d,d*A323411(n/d));

Formula

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

A349400 Dirichlet inverse of EKG-permutation, A064413.

Original entry on oeis.org

1, -2, -4, -2, -3, 7, -12, 8, 6, 7, -15, 18, -14, 41, 3, -12, -16, -4, -22, 9, 63, 33, -30, -49, -26, 28, -10, 15, -39, -2, -32, 6, 103, 13, 30, -69, -19, 31, 67, -68, -44, -218, -23, 39, 36, 70, -52, 38, 88, 67, 65, 52, -55, -21, 20, -294, 147, 69, -66, -52, -31, 35, -144, 48, 16, -240, -37, 93, 165, -180, -76, 78
Offset: 1

Views

Author

Antti Karttunen, Nov 19 2021

Keywords

Links

Crossrefs

Cf. A064413, A323411, A349614.

Programs

  • PARI
    v064413 = readvec("b064413_to.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
memoA349400 = Map();
A349400(n) = if(1==n,1,my(v); if(mapisdefined(memoA349400,n,&v), v, v = -sumdiv(n,d,if(dA064413(n/d)*A349400(d),0)); mapput(memoA349400,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A064413(n/d) * a(d).
Showing 1-5 of 5 results.

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