A143235 -id:A143235 - cp's OEIS frontend

A108461 Table read by antidiagonals: T(n,k) = number of factorizations of (n,k) into pairs (i,j) with i,j>=1, not both 1.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 4, 2, 2, 1, 5, 2, 9, 2, 5, 1, 3, 2, 5, 4, 4, 5, 2, 3, 2, 7, 2, 11, 2, 11, 2, 7, 2, 2, 4, 7, 4, 5, 5, 4, 7, 4, 2, 1, 5, 4, 16, 2, 15, 2, 16, 4, 5, 1, 4, 2, 5, 9, 7, 5, 5, 7, 9, 5, 2, 4, 1, 11, 2, 11, 4, 21, 2, 21, 4, 11, 2, 11, 1, 2, 2, 11, 4, 5, 11, 7

Offset: 1

Christian G. Bower, Jun 03 2005

The rule of building products is (a,b)*(x,y) = (a*x,b*y).

The number of divisors of (n,k) is A143235(n,k)-1, where the subtraction of 1 means that the unit (1,1) is not admitted here. - R. J. Mathar, Nov 30 2017

			1 1 1 2 1 ...
1 2 2 4 2 ...
1 2 2 4 2 ...
2 4 4 9 4 ...
1 2 2 4 2 ...
(6,2)=(6,1)*(1,2)=(3,2)*(2,1)=(3,1)*(2,2)=(1,2)*(6,1), so a(6,2)=5.

R. J. Mathar, Table of n, a(n) for n = 1..1711, the first 59 diagonals.
R. J. Mathar, Java Source code of 2 files to generate the b-file

Cf. A051707, A108455-A108470.
Columns 1-3: A001055, A057567, A057567.
Main diagonal: A108462.

Dirichlet g.f.: A(s, t) = exp(B(s, t)/1 + B(2*s, 2*t)/2 + B(3*s, 3*t)/3 + ...) where B(s, t) = zeta(s)*zeta(t)-1.

A143236 a(n) = A000005(n) * A006218(n).

Original entry on oeis.org

1, 6, 10, 24, 20, 56, 32, 80, 69, 108, 58, 210, 74, 164, 180, 250, 104, 348, 120, 396, 280, 296, 152, 672, 261, 364, 380, 606, 206, 888, 226, 714, 492, 508, 524, 1260, 284, 584, 600, 1264, 320, 1344, 340, 1056, 1092, 744, 376, 1980, 603, 1242, 844, 1302, 438, 1816, 924

Offset: 1

Gary W. Adamson, Aug 01 2008

nonn

			a(4) = 24 = A000005(4) * A006218(4) = 3*8.
a(4) = 24 = sum of row 4 terms of triangle A143235: (3 + 6 + 6 + 9).

Robert Israel, Table of n, a(n) for n = 1..10000

Row sums of triangle A143235.

Cf. A000005, A006218, A143235.

A143236:= func< n | NumberOfDivisors(n)*(&+[Floor(n/k): k in [1..n]]) >;
[A143236(n): n in [1..100] ]; // G. C. Greubel, Sep 12 2024

A143236[n_]:= DivisorSigma[0,n]*Sum[Floor[n/k], {k,n}];
Table[A143236[n], {n,100}] (* G. C. Greubel, Sep 12 2024 *)

A006218(n)=sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2
a(n)=A006218(n)*numdiv(n) \\ Charles R Greathouse IV, Nov 03 2021

from math import isqrt
from sympy import divisor_count
def A143236(n): return (-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))<<1))*divisor_count(n) # Chai Wah Wu, Oct 23 2023

def A143236(n): return sigma(n,0)*sum(int(n//k) for k in range(1,n+1))
[A143236(n) for n in range(1,101)] # G. C. Greubel, Sep 12 2024

a(n) = Sum_{k=1..n} A143235(n,k).

More terms from N. J. A. Sloane, Oct 19 2008

cp's OEIS Frontend

A108461 Table read by antidiagonals: T(n,k) = number of factorizations of (n,k) into pairs (i,j) with i,j>=1, not both 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A143236 a(n) = A000005(n) * A006218(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

SageMath

Formula

Extensions