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A047802 Smallest abundant number (sigma(x) > 2x) which is not divisible by any of the first n primes. 3
12, 945, 5391411025, 20169691981106018776756331, 49061132957714428902152118459264865645885092682687973, 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) exists for every n, since the sum of the inverses of the primes is infinite.

Heuristic: square the first several primes and then add successive primes until the number is abundant. - Fred Schneider, Sep 20 2006

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, Sep 20 2006

a(5) = 13^2 * 17^2 * 19 * 23 * ... * 223 * 227 - Fred Schneider, Sep 20 2006

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

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..5.

Douglas Iannucci, On the smallest abundant number not divisible by the first k primes, Bulletin of the Belgian Mathematical Society 12:1 (2005), pp. 39-44.

FORMULA

Iannucci shows that log a(n) = (n log n)^(2 + o(1)). - Charles R Greathouse IV, Feb 16 2011

EXAMPLE

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.

CROSSREFS

Subsequence of A005101 and A133812; cf. A005231.

Sequence in context: A203599 A114809 A114371 * A180237 A079916 A204327

Adjacent sequences:  A047799 A047800 A047801 * A047803 A047804 A047805

KEYWORD

nonn

AUTHOR

Ulrich Schimke (ulrschimke(AT)aol.com)

EXTENSIONS

2 more terms from Fred Schneider, Sep 20 2006

STATUS

approved

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Last modified November 25 16:38 EST 2014. Contains 249998 sequences.