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 A047802 Smallest abundant number (sigma(x) > 2x) which is not divisible by any of the first n primes. 3

%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 Heuristic: square the first several primes and then add successive primes until the number is abundant. - Fred Schneider (frederick.william.schneider(AT)gmail.com), Sep 20 2006

%C For terms 3 4 and 5, squaring only the first two will be part of the minimal solution: 49061132957714428902152118459264865645885092682687973 = 11^2 * 13^2 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47 * 53 * 59 * 61 * 67 * 71 * 73 * 79 * 83 * 89 * 97 * 101 * 103 * 107 * 109 * 113 * 127 * 131 * 137 - Fred Schneider (frederick.william.schneider(AT)gmail.com), Sep 20 2006

%C a(5) = 13^2 * 17^2 * 19 * 23 * ... * 223 * 227 - Fred Schneider (frederick.william.schneider(AT)gmail.com), Sep 20 2006

%C a(6) = 17^2 * 19^2 * 23^2 * 29 * 31 * ... * 347 * 349 and

%C a(7) = 19^2 * 23^2 * 29^2 * 31 * 37 * ... * 491 * 499 (both coming from D. Iannucci paper). _Michel Marcus_, May 01 2013

%D M. T. Whalen and C. L. Miller, Odd abundant numbers: some interesting observations, Journal of Recreational Mathematics 22 (1990), pp. 257-261.

%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. 39-44.

%F Iannucci shows that log a(n) = (n log n)^(2 + o(1)).

%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; c.f. A005231.

%K nonn,changed

%O 0,1

%A Ulrich Schimke (ulrschimke(AT)aol.com)

%E 2 more terms from Fred Schneider (frederick.william.schneider(AT)gmail.com), Sep 20 2006

%E Reference by _Charles R Greathouse IV_, Dec 07 2009

%E Reference and formula from _Charles R Greathouse IV_, Feb 16 2011

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