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A037992 Smallest number with 2^n divisors.

%I A037992 #120 Nov 29 2024 00:09:03
%S A037992 1,2,6,24,120,840,7560,83160,1081080,17297280,294053760,5587021440,
%T A037992 128501493120,3212537328000,93163582512000,2888071057872000,
%U A037992 106858629141264000,4381203794791824000,188391763176048432000,8854412869274276304000,433866230594439538896000
%N A037992 Smallest number with 2^n divisors.
%C A037992 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
%C A037992 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
%H A037992 Danny Rorabaugh, <a href="/A037992/b037992.txt">Table of n, a(n) for n = 0..350</a> (first 101 terms from T. D. Noe)
%H A037992 Project Euler, <a href="https://projecteuler.net/problem=500">Problem 500!!!</a>
%F A037992 A000005(a(n)) = A000079(n).
%F A037992 a(n) = Product_{k=1..n} A050376(k), product of the first n terms of A050376. - _Lekraj Beedassy_, Jun 30 2004
%F A037992 a(n) = A052330(2^n -1). - _Thomas Ordowski_, Jun 29 2005
%F A037992 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
%F A037992 a(n) = A005179(2^n). - _Ivan N. Ianakiev_, Apr 01 2015
%F A037992 a(n+1)/a(n) = A050376(n+1). - _Jinyuan Wang_, Oct 14 2018
%t A037992 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 *)
%o A037992 (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 */
%o A037992 (Haskell)
%o A037992 a037992 n = head [x | x <- [1..], a000005 x == 2 ^ n]
%o A037992 -- _Reinhard Zumkeller_, Apr 08 2015
%o A037992 (Python)
%o A037992 def a(n):
%o A037992   product = 1
%o A037992   k = 1
%o A037992   for i in range(n+1):
%o A037992     product *= k   # k=A050376(i), for i>=1
%o A037992     while product % k == 0:
%o A037992       k += 1
%o A037992   return product
%o A037992 # _Jason L. Miller_, Mar 20 2024
%Y A037992 Cf. A000005, A000079, A001221, A005179, A007947, A050376, A052330, A074239, A347064.
%K A037992 nonn,nice,easy
%O A037992 0,2
%A A037992 _J. Lowell_
%E A037992 a(18) from _Don Reble_, Aug 20 2002