A077478 -id:A077478 - cp's OEIS frontend

A065608 Sum of divisors of n minus the number of divisors of n.

0, 1, 2, 4, 4, 8, 6, 11, 10, 14, 10, 22, 12, 20, 20, 26, 16, 33, 18, 36, 28, 32, 22, 52, 28, 38, 36, 50, 28, 64, 30, 57, 44, 50, 44, 82, 36, 56, 52, 82, 40, 88, 42, 78, 72, 68, 46, 114, 54, 87, 68, 92, 52, 112, 68, 112, 76, 86, 58, 156, 60, 92, 98, 120, 80, 136, 66, 120, 92

Offset: 1

Jason Earls, Nov 06 2001

Number of permutations p of {1,2,...,n} such that p(k)-k takes exactly two distinct values. Example: a(4)=4 because we have 4123, 3412, 2143 and 2341. - Max Alekseyev and Emeric Deutsch, Dec 22 2006
Number of solutions to the Diophantine equation xy + yz = n, with x,y,z >= 1.
In other words, number of ways to write n = (a + b) * k for positive integers a, b, k. - Gus Wiseman, Mar 25 2021
Not the same as A184396(n): a(66) = 136 while A184396(66) = 137. - Wesley Ivan Hurt, Dec 26 2013
From Gus Wiseman, Mar 25 2021: (Start)
Also the number of compositions of n into an even number of parts with alternating parts equal. These are finite even-length sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i. For example, the a(2) = 1 through a(8) = 11 compositions are:
  (1,1)  (1,2)  (1,3)      (1,4)  (1,5)          (1,6)  (1,7)
         (2,1)  (2,2)      (2,3)  (2,4)          (2,5)  (2,6)
                (3,1)      (3,2)  (3,3)          (3,4)  (3,5)
                (1,1,1,1)  (4,1)  (4,2)          (4,3)  (4,4)
                                  (5,1)          (5,2)  (5,3)
                                  (1,2,1,2)      (6,1)  (6,2)
                                  (2,1,2,1)             (7,1)
                                  (1,1,1,1,1,1)         (1,3,1,3)
                                                        (2,2,2,2)
                                                        (3,1,3,1)
                                                        (1,1,1,1,1,1,1,1)
The odd-length version is A062968.
The version with alternating parts weakly decreasing is A114921, or A342528 if odd-length compositions are included.
The version with alternating parts unequal is A342532, or A224958 if odd-length compositions are included (unordered: A339404/A000726).
Allowing odd lengths as well as even gives A342527.
(End)
Inverse Möbius transform of n-1. - Wesley Ivan Hurt, Jun 29 2024

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
M. Alekseyev, E. Deutsch, and J. H. Steelman, Solution to problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465.
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_2(n).
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Masato Kobayashi, New recurrences for divisor sum functions and triangular numbers, arXiv:2207.05831 [math.NT], 2022.

Cf. A000203, A000005, A134857.
Starting (1, 2, 4, 4, 8, 6, ...), = row sums of triangle A077478. - Gary W. Adamson, Nov 12 2007
Starting with "1" = row sums of triangle A176919. - Gary W. Adamson, Apr 29 2010
Column k=2 of A125182.
A175342/A325545 count compositions with constant/distinct differences.
Cf. A001522, A002843, A008965, A010054, A064410, A064428, A325557, A342495.

List([1..100],n->Sigma(n)-Tau(n)); # Muniru A Asiru, Mar 19 2018

with(numtheory): seq(sigma(n)-tau(n),n=1..70); # Emeric Deutsch, Dec 22 2006

Table[DivisorSigma[1,n]-DivisorSigma[0,n], {n,100}] (* Wesley Ivan Hurt, Dec 26 2013 *)

a(n) = sigma(n) - numdiv(n); \\ Harry J. Smith, Oct 23 2009

from math import prod
from sympy import factorint
def A065608(n):
    f = factorint(n).items()
    return prod((p**(e+1)-1)//(p-1) for p, e in f)-prod(e+1 for p,e in f) # Chai Wah Wu, Jul 16 2022

a(n) = sigma(n) - d(n) = A000203(n) - A000005(n).
a(n) = Sum_{d|n} (d-1). - Wesley Ivan Hurt, Dec 26 2013
G.f.: Sum_{k>=1} x^(2*k)/(1-x^k)^2. - Benoit Cloitre, Apr 21 2003
G.f.: Sum_{n>=1} (n-1)*x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(1-1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018
G.f.: Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 - differentiate equation 1 in Arndt with respect to t, then set x = q and t = q. - Peter Bala, Jan 22 2021
a(n) = A342527(n) - A062968(n). - Gus Wiseman, Mar 25 2021
a(n) = n * A010054(n) - Sum_{k>=1} a(n - k*(k+1)/2), assuming a(n) = 0 for n <= 0 (Kobayashi, 2022). - Amiram Eldar, Jun 23 2023

A051194 Triangular array T read by rows: T(n,k) = number of positive integers that divide both n and k.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2

Offset: 1

Clark Kimberling

The function T(n,k) is defined for all integer n, k but only the values for 1<=k<=n as a triangular array are listed here.

			Triangle begins:
{1};
{1,2};
{1,1,2};
{1,2,1,3};
{1,1,1,1,2};
{1,2,2,2,1,4};
...

Math StackExchange, A question on the discrete Fourier Transform of some function

Cf. A050873 (gcd), A000005 (number of divisors), A077478 (as square array).

Sum of numbers in row n matches A000203. Sum of numbers in first n rows matches A024916.

T[ n_, k_] := Length[Intersection[Divisors @ If[n == 0, 1, n], Divisors @ If[k == 0, 1, k]]] (* Michael Somos, Jul 18 2011 *)

{T(n, k) = sum( i=1, min( abs(n), abs(k)),(n%i == 0) && (k%i == 0))} /* Michael Somos, Jul 18 2011 */

T(n,k) = A000005(A050873(n,k)). - Reinhard Zumkeller, Jun 28 2010

T(n,k) = T(k,n) = T(-n,k) = T(n,-k) = T(n,n+k) = T(n+k,k). - Michael Somos, Jul 18 2011

A159459 Rectangular array read by antidiagonals: a(n,m) = number of divisors of m that don't divide n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 3, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 2, 1, 2, 1, 3, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 0, 0, 1, 1, 1, 2, 0, 2, 2, 2, 1, 4, 1, 0, 1, 0, 2, 0, 2, 1, 3, 1, 2, 1, 4, 1, 3

Offset: 1

Leroy Quet, Apr 12 2009

a(n,1) = 0, for all n. a(1,m) = d(m)-1, for all m.
From Luc Rousseau, Jul 27 2018: (Start)
a(.,m) is periodic with period m.
a(n,m) is the number of nonzero elements S(n) and S(n+m) have in common, where S(n) denotes the set of complex numbers k*(1-exp(i*2*Pi*n/k)), for k positive integer. See illustration, section links.
(End)

			From _Andrew Howroyd_, Jul 27 2018: (Start)
Array begins:
  0 1 1 2 1 3 1 3 2 ...
  0 0 1 1 1 2 1 2 2 ...
  0 1 0 2 1 2 1 3 1 ...
  0 0 1 0 1 2 1 1 2 ...
  0 1 1 2 0 3 1 3 2 ...
  0 0 0 1 1 0 1 2 1 ...
  0 1 1 2 1 3 0 3 2 ...
  0 0 1 0 1 2 1 0 2 ...
  0 1 0 2 1 2 1 3 0 ...
  ...
(End) [corrected by _Jason Yuen_, Feb 05 2025]

Luc Rousseau, illustration, a(n,m) viewed as an intersection of sets of points

Cf. A077478.

A159459 := proc(n,m) numtheory[tau](m)-numtheory[tau](gcd(n,m)) ; end: for d from 2 to 20 do for m from 1 to d-1 do n := d-m ; printf("%d,",A159459(n,m)) ; od: od: # R. J. Mathar, Apr 16 2009

Table[DivisorSigma[0, #] - DivisorSigma[0, GCD[n, #]] &[m - n + 1], {m, 13}, {n, m, 1, -1}] // Flatten (* Michael De Vlieger, Jul 30 2018 *)

\\ port of R.J. Mathar's Maple program
a(n,m)=numdiv(m)-numdiv(gcd(n,m))
for(d=2,20,for(m=1,d-1,n=d-m;print1(a(n,m),", "))) \\ Luc Rousseau, Jul 27 2018

a(n,m) = d(m) - d(gcd(n,m)), where d(m) = A000005(m).

2 terms corrected by R. J. Mathar, Apr 16 2009

cp's OEIS Frontend

A065608 Sum of divisors of n minus the number of divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Python

Formula

A051194 Triangular array T read by rows: T(n,k) = number of positive integers that divide both n and k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A159459 Rectangular array read by antidiagonals: a(n,m) = number of divisors of m that don't divide n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions