A051173 -id:A051173 - cp's OEIS frontend

A071248 a(n) = Product_{k=1..n} lcm(n,k).

1, 4, 54, 768, 75000, 466560, 592950960, 5284823040, 1735643790720, 45360000000000, 1035338990313196800, 102980960177356800, 145077660657859734604800, 154452450072526199193600

Offset: 1

Amarnath Murthy, May 21 2002

nonn

Log(a(n))/n/Log(n) is bounded since n^n < a(n) < n^(2n). It seems that lim n -> infinity Log(a(n))/n/Log(n) exists and = 1.7.... - Benoit Cloitre, Aug 13 2002

T. D. Noe, Table of n, a(n) for n=1..100

Product of terms in n-th row of A051173.

Cf. A067911, A056916, A055774.

A071248 := proc(n) mul( lcm(k,n),k=1..n) ; end: for n from 1 to 10 do printf("%d ",A071248(n)) ; od ; # R. J. Mathar, Apr 03 2007

Table[Product[LCM[k,n],{k,n}],{n,20}] (* Harvey P. Dale, Jun 12 2019 *)

PARI
```
a(n)=prod(k=1,n,lcm(n,k))
```

a(n) = n!*Product_{ d divides n } d^phi(d). - Vladeta Jovovic, Sep 10 2004

a(n) = n!*n^n/A067911(n)=A000142(n)*A000312(n)/A067911(n). - R. J. Mathar, Apr 03 2007

More terms from Benoit Cloitre, Aug 13 2002

A334231 Triangle read by rows: T(n,k) gives the join of n and k in the graded lattice of the positive integers defined by covering relations "n covers (n - n/p)" for all divisors p of n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 6, 4, 5, 5, 15, 5, 5, 6, 6, 6, 6, 15, 6, 7, 7, 7, 7, 35, 7, 7, 8, 8, 12, 8, 10, 12, 14, 8, 9, 9, 9, 9, 45, 9, 21, 18, 9, 10, 10, 15, 10, 10, 15, 35, 10, 45, 10, 11, 11, 33, 11, 11, 33, 77, 11, 99, 11, 11, 12, 12, 12, 12, 15, 12, 14, 12

Offset: 1

Peter Kagey, Apr 19 2020

The poset of the positive integers is defined by covering relations "n covers (n - n/p)" for all divisors p of n.

n appears A332809(n) times in row n.

			The interval [1,15] illustrates that, for example, T(12, 10) = T(6, 5) = 15, T(12, 4) = 12, T(8, 5) = 10, T(3, 1) = 3, etc.
      15
     _/ \_
    /     \
  10       12
  | \_   _/ |
  |   \ /   |
  5    8    6
   \_  |  _/|
     \_|_/  |
       4    3
       |  _/
       |_/
       2
       |
       |
       1
Triangle begins:
  n\k|  1  2  3  4  5  6  7  8  9 10  11 12 13 14
  ---+-------------------------------------------
   1 |  1
   2 |  2  2
   3 |  3  3  3
   4 |  4  4  6  4
   5 |  5  5 15  5  5
   6 |  6  6  6  6 15  6
   7 |  7  7  7  7 35  7  7
   8 |  8  8 12  8 10 12 14  8
   9 |  9  9  9  9 45  9 21 18  9
  10 | 10 10 15 10 10 15 35 10 45 10
  11 | 11 11 33 11 11 33 77 11 99 11  11
  12 | 12 12 12 12 15 12 14 12 18 15  33 12
  13 | 13 13 13 13 65 13 91 13 39 65 143 13 13
  14 | 14 14 14 14 35 14 14 14 21 35  77 14 91 14

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
Mathematics Stack Exchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?
Wikipedia, Semilattice

Cf. A051173, A332809, A333123, A334184, A334230.

\\ This just returns the least (in a normal sense) number x such that both n and k are in its set of descendants:
up_to = 105;
buildWdescsets(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); (v); }
vdescsets = buildWdescsets(100*up_to); \\ XXX - Think about a safe limit here!
A334231tr(n,k) = for(i=max(n,k),oo,if(setsearch(vdescsets[i],n)&&setsearch(vdescsets[i],k),return(i)));
A334231list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, i++; if(i > up_to, return(v)); v[i] = A334231tr(n,k))); (v); };
v334231 = A334231list(up_to);
A334231(n) = v334231[n]; \\ Antti Karttunen, Apr 19 2020

T(n,1) = T(n,n) = n. T(n, 2) = n for n >= 2.
T(x,y) <= lcm(x,y) for any x,y because x is in same chain with lcm(x,y), and y is in same chain with lcm(x,y).
Moreover, empirically it looks like T(x,y) divides lcm(x,y).

A339394 Sum over all partitions of n of the LCM of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 3, 6, 15, 26, 43, 81, 138, 218, 320, 514, 751, 1131, 1570, 2319, 3159, 4457, 6077, 8344, 11224, 15337, 20297, 26908, 35773, 46434, 60711, 78433, 100987, 129222, 166590, 209719, 267120, 335842, 423341, 527739, 659974, 816805, 1015990, 1251686, 1543864

Offset: 0

Alois P. Heinz, Dec 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A003990, A051173, A339312.

b:= proc(n, i, p, d) option remember; `if`(n=0, ilcm(p, d),
      add(b(n-i*j, i-1, p+j, d+signum(j)), j=`if`(i>1, 0..n/i, n)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, LCM[p, d],
     Sum[b[n - i*j, i - 1, p + j, d + Sign[j]],
     {j, If[i > 1, Range[0, n/i], {n}]}]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A242224 Triangular array T read by rows, T(n, k) = nk(gcd(n,k)+2)/gcd(n,k)^2.

Original entry on oeis.org

3, 6, 4, 9, 18, 5, 12, 8, 36, 6, 15, 30, 45, 60, 7, 18, 12, 10, 24, 90, 8, 21, 42, 63, 84, 105, 126, 9, 24, 16, 72, 12, 120, 48, 168, 10, 27, 54, 15, 108, 135, 30, 189, 216, 11, 30, 20, 90, 40, 14, 60, 210, 80, 270, 12, 33, 66, 99, 132, 165, 198, 231, 264, 297, 330, 13

Offset: 1

Michel Marcus, May 08 2014

Described in the CNRS link as a puzzle where op(n,k) is defined by: op(n,n)=n+2, op(n,k)=op(k,n) and op(n,n+k)/op(n,k)=(n+k)/k.

If gcd(n,k)+2 is replaced by gcd(n,k), then triangle A051173 is obtained.

			Triangle begins:
   3;
   6,  4;
   9, 18,  5;
  12,  8, 36,  6;
  15, 30, 45, 60,  7;
  18, 12, 10, 24, 90,  8;
  ...

Indranil Ghosh, Rows 1..100 of triangle, flattened
Ana Rechtman, Mai, 1er défi, Images des Mathématiques, CNRS, 2014 (in French).
See also Comments, Images des Mathématiques, CNRS, 2014.

t[n_, k_] := n*k*(GCD[n, k] +2)/GCD[n, k]^2; Table[ t[n, k], {n, 12}, {k, n}] // Flatten (* Robert G. Wilson v, Jan 21 2018 *)

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(n*k*(gcd(n,k)+2)/gcd(n,k)^2, ", ");); print(););}

A338797 Triangle read by rows: T(n,k) is the least m such that there exist positive integers x, y and z satisfying x/n + y/k = z/m where all fractions are reduced; 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 4, 12, 1, 5, 10, 15, 20, 1, 6, 3, 2, 12, 30, 1, 7, 14, 21, 28, 35, 42, 1, 8, 8, 24, 8, 40, 24, 56, 1, 9, 18, 9, 36, 45, 18, 63, 72, 1, 10, 5, 30, 20, 2, 15, 70, 40, 90, 1, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1

Offset: 1

Peter Kagey, Nov 09 2020

			Table begins:
  n\k|  1   2   3   4   5   6   7   8   9  10   11 12
  ---+-----------------------------------------------
   1 |  1,
   2 |  2,  1,
   3 |  3,  6,  1,
   4 |  4,  4, 12,  1,
   5 |  5, 10, 15, 20,  1,
   6 |  6,  3,  2, 12, 30,  1,
   7 |  7, 14, 21, 28, 35, 42,  1,
   8 |  8,  8, 24,  8, 40, 24, 56,  1,
   9 |  9, 18,  9, 36, 45, 18, 63, 72,  1,
  10 | 10,  5, 30, 20,  2, 15, 70, 40, 90,   1,
  11 | 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,  1,
  12 | 12, 12,  4,  3, 60,  4, 84, 24, 36, 60, 132, 1.
T(20,10) = 4 because 1/20 + 7/10 = 3/4, and there is no choice of numerators on the left that results in a smaller denominator on the right.

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A051173, A051537, A221918.

import Data.Ratio ((%), denominator)
farey n = [k % n | k <- [1..n], gcd n k == 1]
a338797T n k = minimum [denominator $ a + b | a <- farey n, b <- farey k]

A051537(n,k) <= T(n,k) <= A221918(n,k) <= lcm(n,k) = A051173(n,k).

T(n,k) = lcm(n,k) when gcd(n,k) = 1.

A374352 a(n) = [n>1] * a(n-1) + Sum_{d|n} phi(lcm(d,n/d)) where [] is an Iverson bracket.

Original entry on oeis.org

1, 3, 7, 12, 20, 28, 40, 52, 66, 82, 102, 122, 146, 170, 202, 228, 260, 288, 324, 364, 412, 452, 496, 544, 588, 636, 684, 744, 800, 864, 924, 980, 1060, 1124, 1220, 1290, 1362, 1434, 1530, 1626, 1706, 1802, 1886, 1986, 2098, 2186, 2278, 2382, 2472, 2560, 2688

Offset: 1

Alois P. Heinz, Jul 05 2024

nonn

Sum over all positive integers k, m with k*m <= n of phi(lcm(k,m)).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Iverson bracket

Partial sums of A061884.

Cf. A000010, A003990, A051173, A372866.

a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
      a(n-1)+add(phi(ilcm(d, n/d)), d=divisors(n)))
    end:
seq(a(n), n=1..66);

a(n) = Sum_{j=1..n} Sum_{d|j} phi(lcm(d,j/d)).

a(n) = Sum_{j=1..n} A061884(j).

A174554 Smallest k > 2 such that 2|k, 3|k+1, 4|k+2,..., n|k+n-2.

Original entry on oeis.org

4, 8, 14, 62, 62, 422, 842, 2522, 2522, 27722, 27722, 360362, 360362, 360362, 720722, 12252242, 12252242, 232792562, 232792562, 232792562, 232792562, 5354228882, 5354228882, 26771144402, 26771144402, 80313433202, 80313433202

Offset: 2

Michel Lagneau, Mar 22 2010

nonn

We solve the system of n+1 equations : k==2 (mod 2), k==2 (mod 3),...,k==2 (mod n), and then the solutions are k== 2 mod (lcm(2,3,4,...,n)) where lcm(k) is the least common multiple of{1, 2, ..., k}(A003418) .

			a(2) = 4 because 2|4;
a(3) = 8 because 2|8 and 3|9;
a(4) = 14 because 2|14, 3|15 and 4|16;
a(5) = 62 because 2|62, 3|63, 4|64 and 5|65;
a(6) = 62 because 2|62, 3|63, 4|64, 5|65 and 6|66.

Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, arXiv:0906.2295 [math.NT], 2009.
Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, Amer. Math. Monthly, 116 (2009), 836-839.
Eric Weisstein's World of Mathematics, Least Common Multiple

Cf. A002944, A003990, A051173, A000793, A003418, A048691.

with(numtheory):q:=2:for k from 2 to 100 do :q1:= lcm(q,k):q2 :=2+q1 :print(q2): q :=q1 :od :

a(n) = 2 + lcm(2,3,4,...,n) = A003418(n) + 2.

cp's OEIS Frontend

A071248 a(n) = Product_{k=1..n} lcm(n,k).

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A334231 Triangle read by rows: T(n,k) gives the join of n and k in the graded lattice of the positive integers defined by covering relations "n covers (n - n/p)" for all divisors p of n.

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A339394 Sum over all partitions of n of the LCM of the number of parts and the number of distinct parts.

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Mathematica

A242224 Triangular array T read by rows, T(n, k) = n*k*(gcd(n,k)+2)/gcd(n,k)^2.

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A338797 Triangle read by rows: T(n,k) is the least m such that there exist positive integers x, y and z satisfying x/n + y/k = z/m where all fractions are reduced; 1 <= k <= n.

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A374352 a(n) = [n>1] * a(n-1) + Sum_{d|n} phi(lcm(d,n/d)) where [] is an Iverson bracket.

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A174554 Smallest k > 2 such that 2|k, 3|k+1, 4|k+2,..., n|k+n-2.

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Maple

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A242224 Triangular array T read by rows, T(n, k) = nk(gcd(n,k)+2)/gcd(n,k)^2.