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 1-8 of 8 results.

A349615 a(n) = A349613(n) + A349614(n).

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -6, 0, 0, 14, 9, 0, -14, 0, -42, 4, 0, 0, 54, 49, 0, -17, -12, 0, -118, 0, -36, 10, 0, 28, 124, 0, 0, 28, 144, 0, -24, 0, -30, -215, 0, 0, -167, 4, -140, 34, -84, 0, 187, 70, 12, 30, 0, 0, 626, 0, 0, -114, 69, 196, -106, 0, -102, 26, -12, 0, -508, 0, 0, -785, -90, 20, -254, 0, -287, 125
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A349613, A349614.

Programs

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349613(d) * A349614(n/d). [As the sequences are Dirichlet inverses of each other]

A349376 Dirichlet convolution of A006368 with the Dirichlet inverse of A006369, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

Original entry on oeis.org

1, 0, 0, 1, -3, 5, -4, -2, 1, 11, -7, -7, -7, 14, 7, 4, -10, 2, -11, -22, 10, 25, -14, 16, 7, 25, 0, -26, -17, -41, -18, -8, 17, 36, 34, 7, -21, 39, 17, 52, -24, -52, -25, -48, 1, 50, -28, -36, 8, -51, 24, -48, -31, 7, 62, 60, 27, 61, -35, 136, -35, 64, 0, 16, 62, -93, -39, -70, 34, -178, -42, -26, -42, 75, -27, -74
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A006368, A006369, A349368, A349377 (Dirichlet inverse), A349378 (sum with it).
Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

Programs

Formula

a(n) = Sum_{d|n} A006368(d) * A349368(n/d).

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A006368, A006369, A349351, A349376 (Dirichlet inverse), A349378 (sum with it).
Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

Programs

Formula

a(n) = Sum_{d|n} A006369(d) * A349351(n/d).
a(n) = A349378(n) - A349376(n).

A349397 Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.

Original entry on oeis.org

1, 0, 0, 0, 0, -1, 5, -8, 0, 6, 3, -2, 0, -19, 5, 4, 4, -20, 19, -22, -6, 15, -3, 8, 0, 0, -16, -16, 18, -24, 40, -70, -9, 24, -21, 8, 50, -55, -8, 24, -6, 31, 15, -58, -20, 17, 31, -92, -2, -70, 37, 24, 0, 20, 49, 18, -6, -26, 13, -33, 15, -62, -158, -20, 22, -15, 49, -130, 67, 48, 49, -58, 29, -112, -4, 60, -73, -16
Offset: 1

Views

Author

Antti Karttunen, Nov 19 2021

Keywords

Comments

Dirichlet convolution of A064216 with A323893, which is the Dirichlet inverse of A048673. Therefore, convolving A048673 with this sequence gives A064216.
Note how for n = 1 .. 35, a(n) = -A349398(n).

Links

Crossrefs

Cf. A003961, A048673, A064216, A064989, A323893, A349398 (Dirichlet inverse), A349399 (sum with it), A349384.
Cf. also pairs A349376, A349377 and A349613, A349614 for similar constructions.

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A064216(n) = { my(f = factor(n+n-1)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
A349397(n) = sumdiv(n,d,A064216(n/d)*A323893(d));

Formula

a(n) = Sum_{d|n} A064216(n/d) * A323893(d).

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).

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).

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 1-8 of 8 results.

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