A037992 Smallest number with 2^n divisors.

1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 17297280, 294053760, 5587021440, 128501493120, 3212537328000, 93163582512000, 2888071057872000, 106858629141264000, 4381203794791824000, 188391763176048432000, 8854412869274276304000, 433866230594439538896000

Offset: 0

Author

J. Lowell

Keywords

• nonn
• nice
• easy

Comments

Positions where the number of infinitary divisors of n (A037445), increases to a record (cf. A002182), or infinitary analog of highly composite numbers (A002182). - Vladimir Shevelev, May 13-22 2016

Infinitary superabundant numbers: numbers m with record values of the infinitary abundancy index, A049417(m)/m > A049417(k)/k for all k < m. - Amiram Eldar, Sep 20 2019

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..350 (first 101 terms from T. D. Noe)
• Project Euler, Problem 500!!!

Crossrefs

Cf. A000005, A000079, A001221, A005179, A007947, A050376, A052330, A074239, A347064.

Programs

Haskell

a037992 n = head [x | x <- [1..], a000005 x == 2 ^ n]
-- Reinhard Zumkeller, Apr 08 2015

Mathematica

a[0] = 1; a[n_] := a[n] = Catch[ For[ k = 2, True, k++, If[ an = k*a[n-1]; DivisorSigma[0, an] == 2^n, Throw[an]]]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Apr 16 2012 *)

PARI

{a(n)= local(A,m,c,k,p); if(n<1, n==0, c=0; A=1; m=1; while( cMichael Somos, Apr 15 2005 */

Python

def a(n):
  product = 1
  k = 1
  for i in range(n+1):
    product *= k   # k=A050376(i), for i>=1
    while product % k == 0:
      k += 1
  return product
# Jason L. Miller, Mar 20 2024

Formula

A000005(a(n)) = A000079(n).

a(n) = Product_{k=1..n} A050376(k), product of the first n terms of A050376. - Lekraj Beedassy, Jun 30 2004

a(n) = A052330(2^n -1). - Thomas Ordowski, Jun 29 2005

A001221(a(n+1)) <= A001221(a(n))+1, see also A074239; A007947(a(n)) gives a sequence of primorials (A002110) in nondecreasing order. - Reinhard Zumkeller, Apr 16 2006, corrected: Apr 09 2015

a(n) = A005179(2^n). - Ivan N. Ianakiev, Apr 01 2015

a(n+1)/a(n) = A050376(n+1). - Jinyuan Wang, Oct 14 2018

Extensions

a(18) from Don Reble, Aug 20 2002