%I
%S 12,945,5391411025,20169691981106018776756331,
%T 49061132957714428902152118459264865645885092682687973,
%U 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701
%N Smallest abundant number (sigma(x) > 2x) which is not divisible by any of the first n primes.
%C a(n) exists for every n, since the sum of the inverses of the primes is infinite.
%C From _Fred Schneider_, Sep 20 2006; edited by _Danny Rorabaugh_, Nov 26 2018: (Start)
%C Heuristic: Add the squares of several successive primes and then add successive primes until the number is abundant.
%C a(2) = 5^2 * 7 * 11 * 13 * 17 * 19 * 23 * 29;
%C a(3) = 7^2 * 11^2 * 13 * 17 * ... * 61 * 67;
%C a(4) = 11^2 * 13^2 * 17 * 19 * ... * 131 * 137;
%C a(5) = 13^2 * 17^2 * 19 * 23 * ... * 223 * 227. (End)
%C a(6) = 17^2 * 19^2 * 23^2 * 29 * 31 * ... * 347 * 349;
%C a(7) = 19^2 * 23^2 * 29^2 * 31 * 37 * ... * 491 * 499 (both coming from D. Iannucci paper).  _Michel Marcus_, May 01 2013
%C The known terms of this sequence provide Egyptian decompositions of unity in which all the denominators lack the first n primes, as follows: Every term listed in this sequence is a semiperfect number, which means that a subset of its divisors add up to the number itself. The decomposition 1 = 1/a + 1/b + ... + 1/m, where the denominators are a(n) divided by those divisors, is the desired decomposition.  _Javier MÃºgica_, Nov 15 2017
%C a(n) is the product of consecutive primes starting from prime(n+1) raised to nonincreasing powers.  _Jianing Song_, Apr 10 2021
%D M. T. Whalen and C. L. Miller, Odd abundant numbers: some interesting observations, Journal of Recreational Mathematics 22 (1990), pp. 257261.
%H Jeppe Stig Nielsen, <a href="/A047802/b047802.txt">Table of n, a(n) for n = 0..13</a>
%H Thomas Fink, <a href="https://arxiv.org/abs/2008.10398">Recursively abundant and recursively perfect numbers</a>, arXiv:2008.10398 [math.NT], 2020. Mentions this sequence.
%H Douglas Iannucci, <a href="http://projecteuclid.org/euclid.bbms/1113318127">On the smallest abundant number not divisible by the first k primes</a>, Bulletin of the Belgian Mathematical Society 12:1 (2005), pp. 3944.
%F Iannucci shows that log a(n) = (n log n)^(2 + o(1)).  _Charles R Greathouse IV_, Feb 16 2011
%e a(0) = 12, the first abundant number; a(1) = 945, the first odd abundant number; a(5) is the first abundant number not divisible by 2,3,5,7 or 11.
%Y Subsequence of A005101 and A133812; cf. A005231.
%K nonn,changed
%O 0,1
%A Ulrich Schimke (ulrschimke(AT)aol.com)
%E 2 more terms from _Fred Schneider_, Sep 20 2006
