A145444 -id:A145444 - cp's OEIS frontend

A158327 a(n) = A145444(n)-A145511(n).

0, 2, 0, 2, 0, 6, 0, 2, 0, 6, 0, 6, 0, 6, 0, 2, 0, 12, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 18, 0, 2, 0, 6, 0, 12, 0, 6, 0, 6, 0, 18, 0, 6, 0, 6, 0, 6, 0, 12, 0, 6, 0, 20, 0, 6, 0, 6, 0, 18, 0, 6, 0, 2, 0, 18, 0, 6, 0, 18, 0, 12, 0, 6, 0, 6, 0, 18, 0, 6, 0, 6, 0, 18, 0, 6, 0, 6, 0, 36, 0, 6, 0, 6, 0, 6, 0, 12, 0

Offset: 1

Paul Curtz, Mar 16 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A145444, A145511, A158315, A158360.

t1=direuler(p=2, 200, 1/(1-X)^3) ; t2=direuler(p=2, 2, 1+3*X^2+2*X^3, 200) ; a444=dirmul(t1, t2) ; t3=direuler(p=2, 2, 1-2*X^1+7*X^2, 200) ; a511=dirmul(t1, t3); a444-a511 ; print(%) ; \\ R. J. Mathar, Apr 22 2009

More terms from R. J. Mathar, Apr 22 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

Paul Curtz, Mar 27 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

See A145444, A145501, A145511, A158334.

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

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

Paul Curtz, Mar 27 2009

nonn

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

Cf. A112593.

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

N. J. A. Sloane, Mar 14 2009

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

Antti Karttunen, Table of n, a(n) for n = 1..65537
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Acta Cryst. A48 (1992), 500-508. See Table 1, Symmetry Fmmm.

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

Cf. A001620, A082633.

read("transforms") :
nmax := 100 :
L := [1,-2,0,7,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

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 *)

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

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

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)

A145501 Dirichlet g.f.: (1+1/4^s+4/16^s)*zeta(s)^3.

Original entry on oeis.org

1, 3, 3, 7, 3, 9, 3, 13, 6, 9, 3, 21, 3, 9, 9, 25, 3, 18, 3, 21, 9, 9, 3, 39, 6, 9, 10, 21, 3, 27, 3, 43, 9, 9, 9, 42, 3, 9, 9, 39, 3, 27, 3, 21, 18, 9, 3, 75, 6, 18, 9, 21, 3, 30, 9, 39, 9, 9, 3, 63, 3, 9, 18, 67, 9, 27, 3, 21, 9, 27, 3, 78, 3, 9, 18, 21, 9, 27, 3, 75, 15, 9, 3, 63, 9, 9, 9, 39, 3

Offset: 1

N. J. A. Sloane, Mar 14 2009

Dirichlet convolution of [1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,...] with A007425.

Antti Karttunen, Table of n, a(n) for n = 1..10000
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Acta Cryst. A48 (1992), 500-508. See Table 1, Bravais lattice / symmetry Immm.

Cf. A007425, A145444.

Cf. A001620, A082633.

nmax := 100 :
L := [1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

f[p_, e_] := (e + 1)*(e + 2)/2; f[2, 1] = 3; f[2, e_] := 3*(e - 2)*(e - 1) + 7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

up_to = 10000
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=dirmul(t1, t2);
\\ Antti Karttunen, Sep 24 2017, after PARI-code in A145444.

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, if(f[i,2] == 1, 3, 3*(f[i,2]-2)*(f[i,2]-1)+7), (f[i,2]+1)*(f[i,2]+2)/2)); } \\ Amiram Eldar, Oct 25 2022

From Amiram Eldar, Oct 25 2022: (Start):
Multiplicative with a(2) = 3, a(2^e) = 3*(e-2)*(e-1)+7 for e > 1, and a(p^e) = (e+1)*(e+2)/2 if p > 2.
Sum_{k=1..n} a(k) ~ (3/4)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 9*gamma/2 - 3*log(2)/2 - 3/2 and c_2 = 3/2 + 9*gamma*(gamma-1)/2 - 9*gamma*log(2)/2 - 9*gamma_1/2 + 3*log(2)/2 + 5*log(2)^2/2, 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

Paul Curtz, Mar 16 2009

nonn

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

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

Paul Curtz, Mar 17 2009

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

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

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

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

Paul Curtz, Mar 23 2009

nonn

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

Cf. A158280, A145511, A145444.

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

cp's OEIS Frontend

A158327 a(n) = A145444(n)-A145511(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A145501 Dirichlet g.f.: (1+1/4^s+4/16^s)*zeta(s)^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A158315 A158280(n)/7.

Views

Author

Keywords

Comments

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A158647 A145501(16n+8).

Views

Author

Keywords

Comments

Crossrefs

Extensions