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.

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

A323411 Dirichlet inverse of A064664, the inverse permutation of EKG-sequence.

Original entry on oeis.org

1, -2, -5, 1, -10, 16, -14, -4, 19, 31, -20, -21, -28, 43, 89, 4, -33, -98, -37, -42, 125, 61, -43, 48, 76, 85, -87, -58, -57, -409, -61, -1, 179, 100, 255, 203, -67, 112, 251, 98, -74, -573, -81, -85, -559, 130, -89, -100, 146, -370, 296, -107, -100, 548, 347, 145, 332, 172, -107, 846, -115, 184, -783, 3, 506, -825, -128
Offset: 1

Views

Author

Antti Karttunen, Jan 13 2019

Keywords

Links

Crossrefs

Cf. A064413, A064664, A323412.
Cf. also A304526, A304527, A317843, A318664, A318665.

Programs

  • PARI
    up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ From precomputed file.
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);
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA064664(n)));
A323411(n) = v323411[n];

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

A318664 Numerators of the sequence whose Dirichlet convolution with itself yields A064664, the inverse permutation of EKG-sequence.

Original entry on oeis.org

1, 1, 5, 1, 5, -1, 7, 3, -1, -1, 10, 3, 14, -1, -7, 5, 33, 59, 37, 9, -10, -1, 43, -1, -1, -1, 181, 13, 57, 89, 61, 15, -29, -1, -45, 31, 67, -1, -41, 1, 37, 129, 81, 11, 301, -1, 89, 21, 1, 26, -97, 10, 50, -93, -47, -5, -109, -1, 107, -33, 115, -1, 411, 15, -43, 201, 64, 33, -127, 56, 67, 181, 69, -1, 283, 35, -31, 255, 151, 7
Offset: 1

Views

Author

Antti Karttunen, Sep 01 2018

Keywords

Links

Crossrefs

Cf. A064664, A304526, A304527, A305293, A305294, A318665 (denominators).
Cf. also A317929, A317930.

Programs

  • PARI
    v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ From b-file of A064413 prepared beforehand.
A064413(n) = v064413[n];
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
up_to = (2^14);
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&&dA317937.
v318664_65 = DirSqrt(vector(up_to, n, A064664(n)));
A318664(n) = numerator(v318664_65[n]);
A318665(n) = denominator(v318664_65[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064664(n) - Sum_{d|n, d>1, d 1.
For n >= 2, a(2*A000040(n)) = -1.

A318665 Denominators of the sequence whose Dirichlet convolution with itself yields A064664, the inverse permutation of EKG-sequence.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 8, 2, 1, 2, 1, 2, 1, 1, 2, 8, 2, 2, 1, 2, 2, 2, 2, 2, 16, 2, 2, 4, 2, 2, 2, 2, 2, 8, 2, 2, 2, 1, 1, 4, 2, 1, 8, 2, 2, 2, 2, 1, 4, 1, 1, 16, 2, 2, 4, 2, 2, 4, 2, 2, 8, 1, 1, 4, 1, 2, 4, 1, 1, 8, 1, 2, 4, 2, 1, 4, 2, 1, 128, 2, 1, 4, 2, 2, 4, 1, 2, 16, 2, 2, 4, 2, 1, 4, 1, 1, 2, 8, 2, 4, 2, 2, 4
Offset: 1

Views

Author

Antti Karttunen, Sep 01 2018

Keywords

Links

Crossrefs

Cf. A064664, A304526, A304527, A318664 (numerators).

Programs

  • PARI
    v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ From b-file of A064413 prepared previously.
A064413(n) = v064413[n];
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
up_to = (2^14);
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&&dA317937.
v318664_65 = DirSqrt(vector(up_to, n, A064664(n)));
A318664(n) = numerator(v318664_65[n]);
A318665(n) = denominator(v318664_65[n]);

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A064664(n) - Sum_{d|n, d>1, d 1.
Showing 1-5 of 5 results.

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