A134866 -id:A134866 - cp's OEIS frontend

A132442 Triangle whose n-th row consists of the first n terms of the n-th row of A134866.

1, 1, 3, 1, 1, 4, 1, 3, 1, 7, 1, 1, 1, 1, 6, 1, 3, 4, 3, 1, 12, 1, 1, 1, 1, 1, 1, 8, 1, 3, 1, 7, 1, 3, 1, 15, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 3, 1, 3, 6, 3, 1, 3, 1, 18, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 3, 4, 7, 1, 12, 1, 7, 4, 3, 1, 28, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 3, 1, 3, 1, 3, 8, 3, 1, 3, 1, 3, 1, 24

Offset: 1

Gary W. Adamson, Nov 14 2007

Previous name: Triangle, n-th row = first n terms of n-th row of an array formed by A051731 * A127093 (transform).
Right border = sigma(n), A000203.
Row sums = A038040.
The function T(n,k) = T(k,n) is defined for k > n, but only the values of k in 1..n as a triangular array are listed here.

			First few rows of the A134866 array:
  1,  1,  1,  1,  1,  1,  1, ...
  1,  3,  1,  3,  1,  3,  1, ...
  1,  1,  4,  1,  1,  4,  1, ...
  1,  3,  1,  7,  1,  3,  1, ...
  1,  1,  1,  1,  6,  1,  1, ...
  1,  3,  4,  3,  1, 12,  1, ...
  ...
First few rows of the triangle:
  1;
  1,  3;
  1,  1,  4;
  1,  3,  1,  7;
  1,  1,  1,  1,  6;
  1,  3,  4,  3,  1, 12;
  1,  1,  1,  1,  1,  1,  8;
  1,  3,  1,  7,  1,  3,  1, 15;
  ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A051731, A127093.
Cf. A038040 (row sums), A000203 (right border), A050873 (gcd(n,k)).
Cf. A000142 (determinant).
Cf. A134866.

a132442 n k = a132442_tabl !! (n-1) !! (k-1)
a132442_row n = a132442_tabl !! (n-1)
a132442_tabl = map (map a000203) a050873_tabl
-- Reinhard Zumkeller, Dec 12 2015

T[ n_, k_] := If[ n < 1 || k < 1, 0, If[ k > n, T[ k, n], If[ k == 1, 1, If[ n > k, T[ k, Mod[ n, k, 1]],  DivisorSigma [1, n]]]]] (* Michael Somos, Jul 18 2011 *)

{T(n, k) = if( n<1 || k<1, 0, if( k>n, T(k, n), if( k==1, 1, if( n>k, T(k, (n-1)%k+1), sigma( n)))))} /* Michael Somos, Jul 18 2011 */

T(n,k) = A000203(gcd(n,k)). - Reinhard Zumkeller, Dec 12 2015

Missing T(10,9) = 1 inserted by Reinhard Zumkeller, Dec 12 2015

Name edited by Michel Marcus, Dec 21 2022

A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

Original entry on oeis.org

0, 1, 2, 5, 4, 12, 6, 17, 14, 22, 10, 44, 12, 32, 36, 49, 16, 69, 18, 78, 52, 52, 22, 132, 44, 62, 68, 112, 28, 168, 30, 129, 84, 82, 92, 233, 36, 92, 100, 230, 40, 240, 42, 180, 192, 112, 46, 356, 90, 207, 132, 214, 52, 312, 148, 328, 148, 142, 58, 552, 60

Offset: 1

Labos Elemer, May 28 2004

Not all values arise and some arise more than once.
Row sums of triangle A134866. - Gary W. Adamson, Nov 14 2007
Sum of the largest parts of the partitions of n into two parts such that the smaller part divides the larger. - Wesley Ivan Hurt, Dec 21 2017
a(n) is also the sum of all parts minus the total number of parts of all partitions of n into equal parts (an interpretation of the Torlach Rush's formula). - Omar E. Pol, Nov 30 2019
If and only if sigma(n) divides a(n), then n is one of Ore's Harmonic numbers, A001599. - Antti Karttunen, Jul 18 2020

			q^2 + 2*q^3 + 5*q^4 + 4*q^5 + 12*q^6 + 6*q^7 + 17*q^8 + 14*q^9 + ...
For n = 4 the partitions of 4 into equal parts are [4], [2,2], [1,1,1,1]. The sum of all parts is 4 + 2 + 2 + 1 + 1 + 1 + 1 = 12. There are 7 parts, so a(4) = 12 - 7 = 5. - _Omar E. Pol_, Nov 30 2019

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 30.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A001599, A038040, A134866, A152211, A244051, A324121 (= gcd(a(n), sigma(n))).

Cf. A088827 (positions of odd terms).

using AbstractAlgebra
function A094471(n)
    sum(k for k in 0:n if is_divisible_by(n, n - k))
end
[A094471(n) for n in 1:61] |> println  # Peter Luschny, Nov 14 2023

with(numtheory); A094471:=n->n*tau(n)-sigma(n); seq(A094471(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013
divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0):
a := n -> local k; add(`if`(divides(n - k, n), k, 0), k = 0..n):
seq(a(n), n = 1..61);  # Peter Luschny, Nov 14 2023

Table[n*DivisorSigma[0, n] - DivisorSigma[1, n], {n, 1, 100}]

{a(n) = n*numdiv(n) - sigma(n)} /* Michael Somos, Jan 25 2008 */

from math import prod
from sympy import factorint
def A094471(n):
    f = factorint(n).items()
    return n*prod(e+1 for p,e in f)-prod((p**(e+1)-1)//(p-1) for p,e in f)
# Chai Wah Wu, Nov 14 2023

def A094471(n): return sum(k for k in (0..n) if (n-k).divides(n))
print([A094471(n) for n in range(1, 62)])  # Peter Luschny, Nov 14 2023

a(n) = n*tau(n) - sigma(n) = n*A000005(n) - A000203(n). [Previous name.]
If p is prime, then a(p) = p*tau(p)-sigma(p) = 2p-(p+1) = p-1 = phi(p).
If n>1, then a(n)>0.
a(n) = Sum_{d|n} (n-d). - Amarnath Murthy, Jul 31 2005
G.f.: Sum_{k>=1} k*x^(2*k)/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 24 2018
a(n) = A038040(n) - A000203(n). - Torlach Rush, Feb 02 2019

Simpler name by Peter Luschny, Nov 14 2023

A359111 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = sigma(gcd(i,j)).

Original entry on oeis.org

1, 1, 4, 22, 266, 2218, 58100, 644828, 20949776, 502226904, 20622109728, 339816568512, 29770028441472, 568704136553760, 31544507027061120, 1864702918415957568, 150882403284582339072, 3672279699978976000896, 458988841789031457035136, 12369374876487501375431040

Offset: 0

Michel Marcus, Dec 18 2022

nonn

Cf. A000142 (determinant), A134866 (matrix).

a(n) = matpermanent(matrix(n, n, i, j, sigma(gcd(i,j))));

cp's OEIS Frontend

A132442 Triangle whose n-th row consists of the first n terms of the n-th row of A134866.

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A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

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A359111 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = sigma(gcd(i,j)).

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PARI