cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A072938 Highly composite numbers that are half of the next highly composite number.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 2520
Offset: 1

Views

Author

Anonymous, Aug 12 2002

Keywords

Comments

A002182(k) such that A002182(k+1) = 2*A002182(k).
Numbers m such that d(m)>=d(k) for 0A000005(n)}. - Lekraj Beedassy, Dec 16 2004
W. Brefeld (cf. link) gives a proof that there are no other terms. - Klaus Brockhaus, Mar 05 2006

Examples

			6 = A002182(4) is a term since A002182(5) = 12 = 2*6.

Links

Crossrefs

Cf. A002110, A002182, A002201, A003418, A094348, A106037, A165412.

Extensions

Edited by Klaus Brockhaus, Aug 13 2002

A236021 Record values in A236020.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 27720, 30240, 55440, 65520, 83160, 110880, 166320, 196560, 221760, 277200, 332640, 360360, 393120, 720720, 831600, 1441440, 2162160, 2882880
Offset: 1

Views

Author

Jaroslav Krizek, Jan 18 2014

Keywords

Comments

Sequence of numbers from A236020 such that A236020(n) > A236020(k) for all k < n.
A236020 = natural numbers n sorted by increasing values of k(n) = log_tau(n) (sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.
Conjecture: subsequence of A094348.

Crossrefs

Cf. A236020, A236022, A236023, A236024, A236025, A236026.

Extensions

More terms from Jon E. Schoenfield, Nov 12 2016

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.

Original entry on oeis.org

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

Views

Author

Matthew Vandermast, Jun 19 2004

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    (* 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 *)
  • PARI
    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

Formula

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

A140999 Members of A067128 that are the smallest numbers with their prime signatures.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 30, 36, 48, 60, 72, 96, 120, 180, 240, 360, 420, 480, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 4620, 5040, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 25200, 27720, 30240, 36960, 37800, 40320, 45360
Offset: 1

Views

Author

J. Lowell, Jul 28 2008

Keywords

Comments

Includes all members of A002182.
Conjecture (false!): includes all members of A094348.
Contribution from Matthew Vandermast, Oct 10 2008: (Start)
Counterexample to conjecture: 5354228880, the smallest positive multiple of the first 23 positive integers, does not belong to A067128. It is the smallest member of A003418 (a subsequence of A094348) not to be largely composite.
Intersection of A067128 and A025487.
Includes all members of A097212. (End)

Examples

			3 doesn't qualify because it's not the smallest number with its prime signature. 16 does not qualify because it's not a member of A067128.

Links

Programs

  • Mathematica
    PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1};dm=1; Do[d=DivisorSigma[0,n]; If[d>=dm, dm=d; pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]]], {n, 2, 50000}]; ln (* Amiram Eldar, Jun 20 2019 after Robert G. Wilson v at A025487 *)

Extensions

More terms from Matthew Vandermast, Oct 10 2008, Oct 14 2008

A160274 Highly composite numbers A002182(n) with the property that A002182(n+1)/A002182(n) >= A002182(k+1)/A002182(k) for all k>n.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 2520
Offset: 1

Views

Author

Anonymous, May 07 2009

Keywords

Examples

			2520 is a term of this sequence because 2520 is a highly composite number (A002182(18)), A002182(19)/A002182(18) = 2, and 2 >= A002182(k+1)/A002182(k) for all k>18. (In fact, 2 > A002182(k+1)/A002182(k) for all k>18.)

Crossrefs

Cf. A002182, A002201, A072938, A106037, A094348, A003418, A002110.
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