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 1-7 of 7 results.

A304528 Restricted growth sequence transform of A304526, which is Möbius transform of A064664, the inverse of EKG-sequence.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 6, 1, 4, 7, 8, 9, 4, 10, 3, 11, 12, 13, 14, 10, 4, 15, 2, 16, 4, 17, 18, 19, 20, 21, 16, 10, 4, 8, 20, 22, 4, 10, 23, 24, 25, 26, 27, 9, 4, 28, 29, 13, 25, 10, 7, 30, 29, 31, 32, 10, 4, 33, 32, 34, 4, 35, 36, 23, 37, 38, 11, 10, 36, 39, 31, 40, 4, 41, 42, 43, 41, 44, 25, 45, 4, 46, 9, 15, 4, 10, 36, 47, 48, 49, 41, 10, 4, 50, 11, 51
Offset: 1

Views

Author

Antti Karttunen, May 18 2018

Keywords

Links

Crossrefs

Cf. A064664, A304526.

Programs

  • PARI
    \\ Needs also code for A064664 and A304526:
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(1,rgs_transform(vector(32768,n,A304526(n))),"b304528.txt");

A304527 Difference between A064664 (the inverse of EKG-sequence) and its Möbius-transform.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 3, 5, 11, 1, 5, 1, 15, 14, 8, 1, 5, 1, 10, 18, 21, 1, 12, 10, 29, 6, 14, 1, 8, 1, 17, 24, 34, 23, 15, 1, 38, 32, 23, 1, 12, 1, 20, 12, 44, 1, 25, 14, 23, 37, 28, 1, 28, 29, 31, 41, 58, 1, 34, 1, 62, 16, 31, 37, 18, 1, 33, 47, 22, 1, 39, 1, 68, 25, 37, 33, 26, 1, 49, 22, 75, 1, 50, 42, 82, 61, 46, 1, 58, 41, 43, 65, 90, 46, 59, 1
Offset: 1

Views

Author

Antti Karttunen, May 18 2018

Keywords

Links

Crossrefs

Cf. A000040, A008683, A064664, A064955, A304526.

Programs

  • PARI
    A304527(n) = -sumdiv(n, d, (dA064664(d));

Formula

a(n) = A064664(n) - A304526(n).
a(n) = Sum_{d|n, dA304526(d).
a(n) = -Sum_{d|n, dA008683(n/d)*A064664(d).
For n >= 2, a(2*A000040(n))-1 = a(4*A000040(n)) = A064955(n). - Antti Karttunen, Dec 04 2022

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];

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

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A055615, A064413, A064664, A349616 (Dirichlet inverse).
Cf. also A000010, A304526, A349614.

Programs

  • PARI
    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));

Formula

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

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.

A327867 Even bisection of Möbius transform of A064664, the inverse of EKG-sequence.

Original entry on oeis.org

1, 1, -2, 5, -2, 2, -2, 9, 7, 8, -2, 4, -2, 12, 15, 14, -2, 15, -2, 17, 23, 21, -2, 20, 23, 19, 20, 18, -2, 18, -2, 33, 41, 32, 33, 24, -2, 34, 44, 23, -2, 27, -2, 33, 25, 44, -2, 32, 43, 31, 64, 42, -2, 32, 53, 46, 67, 48, -2, 32, -2, 60, 39, 58, 62, 40, -2, 62, 90, 44, -2, 44, -2, 78, 37, 70, 51, 55, -2, 58, 47, 84, -2, 50
Offset: 1

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Links

Crossrefs

Cf. A064413, A064664, A304526.

Formula

a(n) = A304526(2*n).
a(p) = -2 for all odd primes p.
Showing 1-7 of 7 results.

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