A006580 -id:A006580 - cp's OEIS frontend

A003990 Table of lcm(x,y), read along antidiagonals.

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 4, 3, 4, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 12, 5, 12, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 9, 8, 35, 6, 35, 8, 9, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 12, 33, 20, 45, 24

Offset: 1

Marc LeBrun

A(x,x) = x on the diagonal. - Reinhard Zumkeller, Aug 05 2012

			The symmetric array is lcm(x,y) = lcm(y,x):
   1  2  3  4  5  6  7  8  9 10 ...
   2  2  6  4 10  6 14  8 18 10 ...
   3  6  3 12 15  6 21 24  9 30 ...
   4  4 12  4 20 12 28  8 36 20 ...
   5 10 15 20  5 30 35 40 45 10 ...
   6  6  6 12 30  6 42 24 18 30 ...
   7 14 21 28 35 42  7 56 63 70 ...
   8  8 24  8 40 24 56  8 72 40 ...
   9 18  9 36 45 18 63 72  9 90 ...
  10 10 30 20 10 30 70 40 90 10 ...

T. D. Noe, First 100 antidiagonals of array, flattened
Kival Ngaokrajang, Illustration of pattern, where terms with least significant decimal digit equal to zero are colored.
Index entries for sequences related to lcm's

Cf. A003989, A003991, A051173.
A(x, y) = A075174(A003986(A075173(x), A075173(y))) = A075176(A003986(A075175(x), A075175(y))).
Antidiagonal sums are in A006580.
Cf. A002260.

a003990 x y = a003990_adiag x !! (y-1)
a003990_adiag n = a003990_tabl !! (n-1)
a003990_tabl = zipWith (zipWith lcm) a002260_tabl $ map reverse a002260_tabl
-- Reinhard Zumkeller, Aug 05 2012

Table[ LCM[x-y, y], {x, 1, 14}, {y, 1, x-1}] // Flatten (* Jean-François Alcover, Aug 20 2013 *)

A(x,y)=lcm(x,y) \\ Charles R Greathouse IV, Feb 06 2017

A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 4, 3, 4, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 6, 15, 4, 15, 6, 7, 0, 0, 8, 14, 6, 20, 20, 6, 14, 8, 0, 0, 9, 8, 21, 12, 5, 12, 21, 8, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 10, 9, 8, 35, 6, 35, 8, 9, 10, 11, 0

Offset: 0

Mitch Harris, Jun 18 2005

			Seen as an array:
  [0] 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, ...
  [1] 0, 1,  2,  3,  4,  5,  6,  7,  8,  9, ...
  [2] 0, 2,  2,  6,  4, 10,  6, 14,  8, 18, ...
  [3] 0, 3,  6,  3, 12, 15,  6, 21, 24,  9, ...
  [4] 0, 4,  4, 12,  4, 20, 12, 28,  8, 36, ...
  [5] 0, 5, 10, 15, 20,  5, 30, 35, 40, 45, ...
  [6] 0, 6,  6,  6, 12, 30,  6, 42, 24, 18, ...
  [7] 0, 7, 14, 21, 28, 35, 42,  7, 56, 63, ...
  [8] 0, 8,  8, 24,  8, 40, 24, 56,  8, 72, ...
  [9] 0, 9, 18,  9, 36, 45, 18, 63, 72,  9, ...
.
Seen as a triangle T(n, k) = lcm(n - k, k).
  [0] 0;
  [1] 0, 0;
  [2] 0, 1,  0;
  [3] 0, 2,  2,  0;
  [4] 0, 3,  2,  3,  0;
  [5] 0, 4,  6,  6,  4,  0;
  [6] 0, 5,  4,  3,  4,  5, 0;
  [7] 0, 6, 10, 12, 12, 10, 6,  0;
  [8] 0, 7,  6, 15,  4, 15, 6,  7, 0;
  [9] 0, 8, 14,  6, 20, 20, 6, 14, 8, 0;

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows A000027, A109043, A109044, A109045, A109046, A109047, A109048, A109049, A109050, A109051, A109052, A109053, A006580 (row sums of triangle), A001477 (main diagonal, central terms).

Variants: A003990 is (1, 1) based, A051173 (T(n,k) = lcm(n,k)).

T := (n, k) -> ilcm(n - k, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 24 2025

lcm(n, k) = n*k / gcd(n, k) for (n, k) != (0, 0).

A209295 Antidiagonal sums of the gcd(.,.) array A109004.

Original entry on oeis.org

0, 2, 5, 8, 12, 14, 21, 20, 28, 30, 37, 32, 52, 38, 53, 60, 64, 50, 81, 56, 92, 86, 85, 68, 124, 90, 101, 108, 132, 86, 165, 92, 144, 138, 133, 152, 204, 110, 149, 164, 220, 122, 237, 128, 212, 234, 181, 140, 288, 182, 245, 216, 252, 158, 297, 244

Offset: 0

R. J. Mathar, Jan 17 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5000 terms from G. C. Greubel)

Cf. A006580, A018804, A109004.

A209295:= func< n | n eq 0 select 0 else (&+[(n/d+1)*EulerPhi(d): d in Divisors(n)]) >;
[A209295(n): n in [0..40]]; // G. C. Greubel, Jun 24 2024

a:= n-> add(igcd(j, n-j), j=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 25 2019
# Alternative (computes [a(n), n=0..10000] about 25 times faster):
a := n -> add(numtheory:-phi(d)*(n/d + 1), d = numtheory:-divisors(n)):
seq(a(n), n = 0..57); # Peter Luschny, Aug 25 2019

Table[Sum[GCD[n-k,k], {k,0,n}], {n,0,50}] (* G. C. Greubel, Jan 04 2018 *)
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := n + Times @@ f @@@ FactorInteger[n]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Apr 28 2023 *)

a(n) = n + sum(k=1, n, gcd(n,k)); \\ Michel Marcus, Jan 05 2018

def A209295(n): return sum((n/k+1)*euler_phi(k) for k in (1..n) if (k).divides(n))
[A209295(n) for n in range(41)] # G. C. Greubel, Jun 24 2024

a(0) = 0; a(n) = A018804(n) + n for n > 0. [Amended by Georg Fischer, Jan 25 2020]

a(n) = Sum_{d|n} phi(d)*(n/d + 1) for n >= 1. - Peter Luschny, Aug 25 2019

A082022 In the following square array a(i,j) = Least Common Multiple of i and j. Sequence contains the product of the terms of the n-th antidiagonal.

Original entry on oeis.org

1, 4, 18, 576, 1200, 518400, 1587600, 180633600, 1646023680, 13168189440000, 461039040000, 229442532802560000, 86553098311680000, 3753113311877529600, 834966920275488000000

Offset: 1

Amarnath Murthy, Apr 06 2003

nonn

If n is even and n+1 is prime, a(n) = n^2 * (n-1)!^2. If n is odd and >3, 2*(n+1)*a(n) is a perfect square, the root of which has the factor 1/2*n*(n-1)*((n-1)/2)!. This was proved by Lawrence Sze. - Ralf Stephan, Nov 16 2004

			1 2 3 4 5...
2 2 6 4 10...
3 6 3 12 15...
4 4 12 4 20...
5 10 15 20 5...
...
The same array in triangular form is
1
2 2
3 2 3
4 6 6 4
5 4 3 4 5
...
Sequence contains the product of the terms of the n-th row.

Cf. A006580, A003990, A082292.

Equals A001044(n) / A051190(n+1).

for(n=1,20,p=1:for(k=1,n,p=p*lcm(k,n+1-k)):print1(p","))

Prod(k=1...n, lcm(k, n+1-k)).

Corrected and extended by Ralf Stephan, Apr 08 2003

A338798 a(n) = Sum_{k=1..n-1} lcm(lcm(n, k), lcm(n, n-k)).

Original entry on oeis.org

0, 2, 12, 28, 100, 90, 392, 408, 792, 810, 2420, 1356, 4732, 3346, 4560, 6320, 13872, 7506, 21660, 12140, 18900, 21802, 46552, 22008, 53000, 43290, 61668, 49980, 117740, 48450, 153760, 100192, 123552, 129506, 169260, 111420, 312132, 203642, 245544, 195640

Offset: 1

Sebastian Karlsson, Jan 18 2021

nonn

Sebastian Karlsson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000010, A006580, A023900, A051193, A057661, A138421.

a[n_] := Sum[LCM[LCM[n, k], LCM[n, n - k]], {k, 1, n - 1}];
Table[a[n], {n, 1, 40}] (* Robert P. P. McKone, Jan 18 2021 *)

a(n) = sum(k=1, n-1, lcm(lcm(n, k), lcm(n, n-k))); \\ Michel Marcus, Jan 18 2021

from math import gcd
for n in range(1, 41):
    print(n*sum([k*(n-k)//(gcd(n,k)**2) for k in range(1, n)]), end=', ')

a(n) = n*Sum_{k=1..n-1} k*(n-k)/gcd(n,k)^2.
a(n) = (1/6)*n*Sum_{d|n} d*(d*phi(d) - A023900(d)).
a(p^e) = (1/6)*p^(e+1)*(p^e-1)*(p^(e+1) + p^(2*e+1) + p^2 + 2*p + 1)/(p^2 + p + 1).
a(prime(n)) = A138421(n). - Michel Marcus, Jan 20 2021

cp's OEIS Frontend

A003990 Table of lcm(x,y), read along antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A209295 Antidiagonal sums of the gcd(.,.) array A109004.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A082022 In the following square array a(i,j) = Least Common Multiple of i and j. Sequence contains the product of the terms of the n-th antidiagonal.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A338798 a(n) = Sum_{k=1..n-1} lcm(lcm(n, k), lcm(n, n-k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula