A096180 -id:A096180 - cp's OEIS frontend

A096179 Triangle read by rows: T(n,k) is the smallest positive integer having at least k of the first n positive integers as divisors.

1, 1, 2, 1, 2, 6, 1, 2, 4, 12, 1, 2, 4, 12, 60, 1, 2, 4, 6, 12, 60, 1, 2, 4, 6, 12, 60, 420, 1, 2, 4, 6, 12, 24, 120, 840, 1, 2, 4, 6, 12, 24, 72, 360, 2520, 1, 2, 4, 6, 12, 24, 60, 120, 360, 2520, 1, 2, 4, 6, 12, 24, 60, 120, 360, 2520, 27720, 1, 2, 4, 6, 12, 12, 24, 60, 120, 360

Offset: 1

Matthew Vandermast, Jun 19 2004

			Triangle begins:
1
1 2
1 2 6
1 2 4 12
1 2 4 12 60
1 2 4  6 12 60

Wikipedia, Table of divisors.

Main diagonal is A003418. Minimum in column k is A061799(k). See also A094348, A096180.

with(combstruct):
a096179_row := proc(n) local k,L,l,R,LCM,comb;
R := NULL; LCM := ilcm(seq(i,i=[$1..n]));
for k from 1 to n-1 do
   L := LCM;
   comb := iterstructs(Combination(n),size=k):
   while not finished(comb) do
      l := nextstruct(comb);
      L := min(L,ilcm(op(l)));
   od;
   R := R,L;
od;
R,LCM end; # Peter Luschny, Dec 06 2010

(* Triangular *)
A096179[n_,k_]:=Min[LCM@@@Subsets[Range[n],{k}]];
A002024[n_]:=Floor[1/2+Sqrt[2*n]];
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
(* Linear *)
A096179[n_]:=A096179[n]=A096179[A002024[n],A002260[n]];
(* Enrique Pérez Herrero_, Dec 08 2010 *)

A096179(n,k)={ my(m=lcm(vector(k,i,i))); forvec(v=vector(k-1,i,[2,n]), m>lcm(v) & m=lcm(v), 2); m } \\ M. F. Hasler, Nov 30 2010

T(n,k) = min { lcm(x_1,...,x_k) ; 0 < x_1 < ... < x_k <= n }

A027457 a(n) = (H(n) - 1)*lcm{1,...,n}, where H(n) is the n-th harmonic number.

Original entry on oeis.org

0, 1, 5, 13, 77, 87, 669, 1443, 4609, 4861, 55991, 58301, 785633, 811373, 835397, 1715839, 29889983, 30570663, 593094837, 604734465, 615819825, 626401305, 14640022575, 14863115445, 75386423001, 76416082401, 232222818803, 235091155703, 6897956948587

Offset: 1

Olivier Gérard

nonn

Second column of A027446. - Olivier Gérard, Dec 11 1999

Rows sums of (A002262*A096180). - Eric Desbiaux, Apr 23 2013

			a(3) = (1/2+1/3)*lcm(2,3) = 5.

Robert Israel, Table of n, a(n) for n = 1..2296

Cf. A001008, A002805, A003418.

[(HarmonicNumber(n)-1)*Lcm([1..n]): n in [1..30]]; // Vincenzo Librandi, Dec 14 2016

A027457 := n -> (Psi(n+1)-1+gamma)*lcm(seq(k,k=1..n)): # Peter Luschny, Dec 01 2011
# alternative:
A[1]:= 0: L[1]:= 1:
for n from 1 to 50 do
   L[n+1]:= ilcm(L[n],n+1);
   A[n+1]:= L[n+1]*(A[n]/L[n] + 1/(n+1))
od:
seq(A[n],n=1..50); # Robert Israel, Dec 14 2016

a[n_] := (HarmonicNumber[n] - 1)*LCM @@ Range[n]; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Mar 05 2013 *)

a(n) = (sum(i=1, n, 1/i)-1)*lcm([1..n]); \\ Michel Marcus, Jul 23 2022

Numerators of sequence a[ 2, n ] in (a[ i, j ])^2 where a[ i, j ] = 1/i if j<=i, 0 if j>i. - N. J. A. Sloane, Feb 24 2006
a(n) = (Psi(n+1)-1+gamma)*LCM(n), LCM(n) = lcm{1..n}. - Peter Luschny, Dec 01 2011
a(n+1) = A003418(n+1)*(a(n)/A003418(n)+1/(n+1)). - Robert Israel, Dec 14 2016

New name, offset changed to 1, a(1) and a(21)-a(29) added. - Peter Luschny, Dec 01 2011

A096294 Triangle T(n,k) read by rows: for n >=0 and n >= k >=0, the fraction of positive integers with exactly k of the first n primes as divisors is T(n,k)/A002110(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 8, 14, 7, 1, 48, 92, 56, 13, 1, 480, 968, 652, 186, 23, 1, 5760, 12096, 8792, 2884, 462, 35, 1, 92160, 199296, 152768, 54936, 10276, 1022, 51, 1, 1658880, 3679488, 2949120, 1141616, 239904, 28672, 1940, 69, 1, 36495360, 82607616

Offset: 0

Matthew Vandermast, Jun 24 2004

Sum of entries in n-th row is A002110(n), the product of the first n primes (primorial numbers, first definition).
From Peter Munn, Apr 10 2017: (Start)
T(n,k) is a count of those integers in any interval of A002110(n) integers that have exactly k of the first n primes as divisors. The count is the same for each such interval because each of the first n primes is a factor of an integer m if and only if it is a factor of m + A002110(n).
A284411(m) is least p=prime(n) such that 2*Sum_{k=0..m-1} T(n,k) < A002110(n).
(End)

			Triangle begins:
1
1 1
2 3 1
8 14 7 1
48 92 56 13 1
480 968 652 186 23 1

First column is A005867; second column is A078456. See also A096180.

Cf. A194156, A284411.

primo(n) = prod(k=1, n, prime(k));
row(n) = {v = vector(n+1); for (k=1, primo(n), f = factor(k)[,1]; v[1+sum(j=1, #f, primepi(f[j])<=n)]++;); v;} \\ Michel Marcus, Apr 29 2017

A217863 a(n) = phi(lcm(1,2,3,...,n)), where phi is Euler's totient function.

Original entry on oeis.org

1, 1, 2, 4, 16, 16, 96, 192, 576, 576, 5760, 5760, 69120, 69120, 69120, 138240, 2211840, 2211840, 39813120, 39813120, 39813120, 39813120, 875888640, 875888640, 4379443200, 4379443200, 13138329600, 13138329600, 367873228800, 367873228800, 11036196864000

Offset: 1

Joshua S.M. Weiner, Oct 13 2012

This is a composition f(g(x)). g(x) = lcm(1...x) and f(x) = phi(x), Euler's totient function. The sequence generated is the number of prime congruence classes (prime spokes) for wheel factorization in mod g(x).

First column of A096180. - Eric Desbiaux, Apr 23 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000010 (Euler phi), A003418 (LCM), A072211, A173557.

a217863 = a000010 . a003418  -- Reinhard Zumkeller, Nov 24 2012

with(numtheory): a:=n->phi(lcm(seq(m,m=1..n))): seq(a(n),n=1..40); # Muniru A Asiru, Feb 20 2019

EulerPhi[Table[LCM @@ Range[n], {n, 35}]] (* T. D. Noe, Oct 16 2012 *)

a(n) = eulerphi(lcm(vector(n, k, k))); \\ Michel Marcus, Aug 25 2015

a(n) = A000010(A003418(n)). - Omar E. Pol, Nov 25 2012
From Peter Bala, Feb 19 2019: (Start)
a(n) = Product_{k = 1..n} A072211(k).
With p denoting a prime, a(n) = ( Product_{p <= n} (p - 1) ) * ( Product_{p^2 <= n} p ) * ( Product_{p^3 <= n} p ) * ... . For example, a(16) = ((2-1)*(3-1)*(5-1)*(7-1)*(11-1)*(13-1)) * (2*3) * 2 * 2 = 138240. (End)

A138553 Table read by rows: T(n,k) is the number of divisors of k that are <= n.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Mar 24 2008

Suggested by a question from Eric Desbiaux.
The row lengths are the lengths before the pattern for n repeats.
Antidiagonal sums A070824. [From Eric Desbiaux, Dec 10 2009]

			The first few rows start:
1, [A000012]
1, 2, [A000034]
1, 2, 2, 2, 1, 3, [A083039]
1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, [A083040]

Row lengths A003418, row sums A025529, frequencies in rows A096180.

Cf. A243987

lista(nrows) = {for (n=1, nrows, for (k=1, lcm(vector(n, i, i)), print1(sumdiv(k, d, d <=n), ", ");); print(););} \\ Michel Marcus, Jun 19 2014

T(n,k) = sum_{i|k, i<=n} 1.

Definition corrected by Franklin T. Adams-Watters, Jun 19 2014

cp's OEIS Frontend

A096179 Triangle read by rows: T(n,k) is the smallest positive integer having at least k of the first n positive integers as divisors.

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A027457 a(n) = (H(n) - 1)*lcm{1,...,n}, where H(n) is the n-th harmonic number.

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A096294 Triangle T(n,k) read by rows: for n >=0 and n >= k >=0, the fraction of positive integers with exactly k of the first n primes as divisors is T(n,k)/A002110(n).

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A217863 a(n) = phi(lcm(1,2,3,...,n)), where phi is Euler's totient function.

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A138553 Table read by rows: T(n,k) is the number of divisors of k that are <= n.

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