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.

A158280 Octosection: A145511(8n+4) or A145501(8n+4).

Original entry on oeis.org

7, 21, 21, 21, 42, 21, 21, 63, 21, 21, 63, 21, 42, 70, 21, 21, 63, 63, 21, 63, 21, 21, 126, 21, 42, 63, 21, 63, 63, 21, 21, 126, 63, 21, 63, 21, 21, 126, 63, 21, 105, 21, 63, 63, 21, 63, 63, 63, 21, 126, 21, 21, 189, 21, 21, 63, 21, 63, 126, 63, 42, 63, 70, 21, 63, 21, 63, 210, 21, 21
Offset: 0

Views

Author

Paul Curtz, Mar 15 2009

Keywords

Comments

All entries are multiples of 7, cf. A158315.

Extensions

Edited and extended by R. J. Mathar, Apr 04 2009

A158360 a(n) = A145511(n) - A145501(n).

Original entry on oeis.org

0, -2, 0, 0, 0, -6, 0, 6, 0, -6, 0, 0, 0, -6, 0, 12, 0, -12, 0, 0, 0, -6, 0, 18, 0, -6, 0, 0, 0, -18, 0, 18, 0, -6, 0, 0, 0, -6, 0, 18, 0, -18, 0, 0, 0, -6, 0, 36, 0, -12, 0, 0, 0, -20, 0, 18, 0, -6, 0, 0, 0, -6, 0, 24, 0, -18, 0, 0, 0, -18, 0, 36, 0, -6, 0, 0, 0, -18, 0, 36, 0, -6, 0, 0, 0, -6, 0
Offset: 1

Views

Author

Paul Curtz, Mar 17 2009

Keywords

Links

Crossrefs

Cf. A145444, A145501, A145511, A158327, A158801.

Programs

  • PARI
    up_to = 65537;
t1 = direuler(p=2, up_to, 1/(1-X)^3);
t2 = direuler(p=2, 2, 1+1*X^2+4*X^4, up_to);
t3 = direuler(p=2, 2, 1-2*X^1+7*X^2, up_to);
v145501 = dirmul(t1, t2);
v145511 = dirmul(t1, t3);
A158360(n) = (v145511[n] - v145501[n]); \\ (after code in A145501 and A145511) - Antti Karttunen, Sep 27 2018

Extensions

Edited and extended by R. J. Mathar, Apr 08 2009

A158801 a(n) = A145444(n) - A145501(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 0, 6, 0, 0, 0, 14, 0, 0, 0, 6, 0, 0, 0, 24, 0, 0, 0, 6, 0, 0, 0, 20, 0, 0, 0, 12, 0, 0, 0, 24, 0, 0, 0, 6, 0, 0, 0, 42, 0, 0, 0, 6, 0, 0, 0, 24, 0, 0, 0, 18, 0, 0, 0, 26, 0, 0, 0, 6, 0, 0, 0, 48, 0, 0, 0, 6, 0, 0, 0, 42, 0, 0, 0, 18, 0, 0, 0, 24, 0, 0, 0, 6, 0, 0, 0, 60, 0, 0, 0
Offset: 1

Views

Author

Paul Curtz, Mar 27 2009

Keywords

Links

Crossrefs

See A145444, A145501, A145511, A158334.

Programs

  • PARI
    up_to = 1001;
t1=direuler(p=2, up_to, 1/(1-X)^3);
t2=direuler(p=2, 2, 1+3*X^2+2*X^3, up_to);
t3=dirmul(t1, t2); \\ For A145444
u1=direuler(p=2, up_to, 1/(1-X)^3);
u2=direuler(p=2, 2, 1+1*X^2+4*X^4, up_to);
u3=dirmul(u1, u2); \\ For A145501
A158801(n) = (t3[n]-u3[n]); \\ Antti Karttunen, Jul 21 2018

Extensions

Edited and extended by R. J. Mathar, Apr 08 2009

A158647 A145501(16n+8).

Original entry on oeis.org

13, 39, 39, 39, 78, 39, 39, 117, 39, 39, 117, 39, 78, 130, 39, 39, 117, 117, 39, 117, 39, 39, 234, 39, 78, 117, 39, 117, 117, 39, 39, 234, 117, 39, 117, 39, 39, 234, 117, 39, 195, 39, 117, 117, 39, 117, 117, 117, 39, 234, 39, 39, 351, 39, 39, 117, 39, 117, 234, 117, 78, 117, 130
Offset: 0

Views

Author

Paul Curtz, Mar 23 2009

Keywords

Comments

Apparently this is the same as 13*A158315(n).

Crossrefs

Cf. A158280, A145511, A145444.

Extensions

Edited and extended by R. J. Mathar, Apr 04 2009

A158805 a(3n) = A145511(n+1). a(3n+1) = A145501(n+1). a(3n+2) =A145444(n+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 3, 7, 7, 9, 3, 3, 3, 3, 9, 9, 3, 3, 3, 19, 13, 21, 6, 6, 6, 3, 9, 9, 3, 3, 3, 21, 21, 27, 3, 3, 3, 3, 9, 9, 9, 9, 9, 37, 25, 39, 3, 3, 3, 6, 18, 18, 3, 3, 3, 21, 21, 27, 9, 9, 9, 3, 9, 9, 3, 3, 3, 57, 39, 63, 6, 6, 6, 3, 9, 9, 10, 10, 10, 21, 21, 27, 3, 3, 3, 9, 27, 27, 3, 3, 3, 61
Offset: 0

Views

Author

Paul Curtz, Mar 27 2009

Keywords

Comments

Why would one clump the three series in this way? [R. J. Mathar, Apr 08 2009]

Crossrefs

Cf. A112593.

Extensions

Edited and extended by R. J. Mathar, Apr 08 2009

A145511 Dirichlet g.f.: (1-2/2^s+7/4^s)*zeta(s)^3.

Original entry on oeis.org

1, 1, 3, 7, 3, 3, 3, 19, 6, 3, 3, 21, 3, 3, 9, 37, 3, 6, 3, 21, 9, 3, 3, 57, 6, 3, 10, 21, 3, 9, 3, 61, 9, 3, 9, 42, 3, 3, 9, 57, 3, 9, 3, 21, 18, 3, 3, 111, 6, 6, 9, 21, 3, 10, 9, 57, 9, 3, 3, 63, 3, 3, 18, 91, 9, 9, 3, 21, 9, 9, 3, 114, 3, 3, 18, 21, 9, 9, 3, 111, 15, 3, 3, 63, 9, 3, 9, 57, 3, 18, 9
Offset: 1

Views

Author

N. J. A. Sloane, Mar 14 2009

Keywords

Comments

Dirichlet convolution of [1,-2,0,7,0,0,0,0,...] and A007425. - R. J. Mathar, Feb 07 2011

Links

Crossrefs

Cf. A007425, A145399, A145444, A145501, A158327, A158360.
Cf. A001620, A082633.

Programs

  • Maple
    read("transforms") :
nmax := 100 :
L := [1,-2,0,7,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017
  • Mathematica
    f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := 3*(e - 1)*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)
  • PARI
    up_to = 65537;
t1 = direuler(p=2, up_to, 1/(1-X)^3);
t3 = direuler(p=2, 2, 1-2*X^1+7*X^2, up_to);
v145511 = dirmul(t1, t3);
A145511(n) = v145511[n]; \\ Antti Karttunen, Sep 27 2018, after R. J. Mathar's PARI-code for A158327
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*(f[i,2]-1)*f[i,2]+1, (f[i,2]+1)*(f[i,2]+2)/2)); } \\ Amiram Eldar, Oct 25 2022

Formula

From Amiram Eldar, Oct 25 2022: (Start)
Multiplicative with a(2^e) = 3*(e-1)*e+1 and a(p^e) = (e+1)*(e+2)/2 if p > 2.
Sum_{k=1..n} a(k) ~ (7/8)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 21*gamma/4 - 5*log(2)/2 - 7/4 and c_2 = 7/4 + 21*gamma*(gamma-1)/4 - 15*gamma*log(2)/2 - 21*gamma_1/4 + 5*log(2)/2 + 3*log(2)^2, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)

A145444 Dirichlet g.f.: (1+3/4^s+2/8^s)*zeta(s)^3.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 3, 21, 6, 9, 3, 27, 3, 9, 9, 39, 3, 18, 3, 27, 9, 9, 3, 63, 6, 9, 10, 27, 3, 27, 3, 63, 9, 9, 9, 54, 3, 9, 9, 63, 3, 27, 3, 27, 18, 9, 3, 117, 6, 18, 9, 27, 3, 30, 9, 63, 9, 9, 3, 81, 3, 9, 18, 93, 9, 27, 3, 27, 9, 27, 3, 126, 3, 9, 18, 27, 9, 27, 3, 117, 15, 9, 3, 81, 9, 9, 9, 63
Offset: 1

Views

Author

N. J. A. Sloane, Mar 14 2009

Keywords

Comments

Dirichlet convolution of [1,0,0,3,0,0,0,2,0,0,...] with A007425. - R. J. Mathar, Sep 25 2017

Links

Crossrefs

Cf. A007425, A145501.
Cf. A001620, A082633.

Programs

  • Maple
    nmax := 10000 :
L := [1,0,0,3,0,0,0,2,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017
  • Mathematica
    f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := 3*(e - 1)*e + 3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)
  • PARI
    t1=direuler(p=2,200,1/(1-X)^3)
t2=direuler(p=2,2,1+3*X^2+2*X^3,200)
t3=dirmul(t1,t2)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*(f[i,2]-1)*f[i,2]+3, (f[i,2]+1)*(f[i,2]+2)/2)); } \\ Amiram Eldar, Oct 25 2022

Formula

From Amiram Eldar, Oct 25 2022: (Start):
Multiplicative with a(2^e) = 3*(e-1)*e+3 for e > 0, and a(p^e) = (e+1)*(e+2)/2 if p > 2.
Sum_{k=1..n} a(k) ~ n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 6*gamma - 9*log(2)/4 - 2 and c_2 = 2 + 6*gamma*(gamma-1) - 27*gamma*log(2)/4 - 6*gamma_1 + 9*log(2)/4 + 21*log(2)^2/8, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)

A158315 A158280(n)/7.

Original entry on oeis.org

1, 3, 3, 3, 6, 3, 3, 9, 3, 3, 9, 3, 6, 10, 3, 3, 9, 9, 3, 9, 3, 3, 18, 3, 6, 9, 3, 9, 9, 3, 3, 18, 9, 3, 9, 3, 3, 18, 9, 3, 15, 3, 9, 9, 3, 9, 9, 9, 3, 18, 3, 3, 27, 3, 3, 9, 3, 9, 18, 9, 6, 9, 10, 3, 9, 3, 9, 30, 3, 3, 9, 9, 9, 18, 3, 3, 18, 9, 3, 9, 9, 3, 27, 3, 6, 18, 3, 18, 9, 3, 3, 9, 9, 9, 30, 3, 3, 27
Offset: 0

Views

Author

Paul Curtz, Mar 16 2009

Keywords

Comments

A kind of core or backbone of A145511, A145501 and A145444.

Extensions

Edited and extended by R. J. Mathar, Apr 04 2009
Showing 1-8 of 8 results.

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