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A019505 a(1)=1; for n > 1, a(n) is the smallest number with the same number of divisors as 2*a(n-1).

%I A019505 #57 Jan 28 2022 17:48:07
%S A019505 1,2,4,6,12,24,48,60,120,240,360,720,1260,2520,5040,10080,20160,27720,
%T A019505 55440,110880,221760,332640,665280,1081080,2162160,4324320,8648640,
%U A019505 17297280,21621600,43243200,73513440,147026880,294053760,367567200,735134400,1396755360,2793510720
%N A019505 a(1)=1; for n > 1, a(n) is the smallest number with the same number of divisors as 2*a(n-1).
%C A019505 From _J. Lowell_, Mar 19 2012 and Apr 05 2012: (Start)
%C A019505 Conjectures:
%C A019505 Subsequence of A002182. [This conjecture is false. The 64th term is 97039187544499200, which has exactly 63360 divisors, but is NOT in A002182; which has the smaller number 74801040398884800, which has 64512 divisors. - _J. Lowell_, Nov 29 2021]
%C A019505 In order for n to be followed by a number less than 2n, a requirement is that the number of 2's in the prime factorization of n must not be of the form p-2 where p is a prime.
%C A019505 There are infinitely many values where n, 2n, and 3n are all in this sequence. (It can be proved that n, 2n, 3n, and 4n can never all be in this sequence.)
%C A019505 In any group of 3 consecutive terms of this sequence a,b,c at most one of the following statements is true:
%C A019505 The value of b is less than twice a.
%C A019505 The value of c is less than twice b. [This conjecture is false. Terms 121-123 are 9363553722094352358689983872000, 14258138622280036546187020896000, and 26139920807513400334676204976000. - _J. Lowell_, Jan 28 2022]
%C A019505 There are terms divisible by 2^k no matter how large k is.
%C A019505 The primes and the powers of 3 are the only numbers that never "come and go" from the lists of divisors of the numbers in the sequence as the terms advance. [This conjecture is false. The 122nd term, 14258138622280036546187020896000, is a multiple of 3^5=243, but the 123rd term, 26139920807513400334676204976000, is not. - _J. Lowell_, Jan 28 2022]
%C A019505 (End)
%H A019505 David A. Corneth, <a href="/A019505/b019505.txt">Table of n, a(n) for n = 1..172</a> (first 63 terms from R. J. Mathar)
%H A019505 Michael De Vlieger, <a href="/A019505/a019505.txt">Concordance of a(n), A002182, and A025487</a>, n = 1..176, giving indices in A002182 and A025487 and number of divisors of a(n).
%F A019505 a(n) = A140635(2*a(n-1)). - _J. Lowell_, May 20 2008
%e A019505 After a(3)=4 we argue as follows: 2*4 = 8 has 4 factors (1,2,4,8), but smallest number with 4 factors is 6, so a(4)=6.
%Y A019505 Cf. A020697, A140635.
%K A019505 nonn
%O A019505 1,2
%A A019505 _J. Lowell_