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A037992 Smallest number with 2^n divisors. 38
1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 17297280, 294053760, 5587021440, 128501493120, 3212537328000, 93163582512000, 2888071057872000, 106858629141264000, 4381203794791824000, 188391763176048432000, 8854412869274276304000, 433866230594439538896000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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!!!
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
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 *)
PROG
(PARI) {a(n)= local(A, m, c, k, p); if(n<1, n==0, c=0; A=1; m=1; while( c<n, m++; if( isprime(m) || ( (k=ispower(m, , &p)) && isprime(p)& k ==2^valuation(k, 2) ), A*=m; c++)); A)} /* Michael Somos, Apr 15 2005 */
(Haskell)
a037992 n = head [x | x <- [1..], a000005 x == 2 ^ n]
-- Reinhard Zumkeller, Apr 08 2015
CROSSREFS
Sequence in context: A024923 A334395 A347064 * A335386 A114779 A317681
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
a(18) from Don Reble, Aug 20 2002
STATUS
approved

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Last modified February 21 20:41 EST 2024. Contains 370237 sequences. (Running on oeis4.)