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

12, 945, 5391411025, 20169691981106018776756331, 49061132957714428902152118459264865645885092682687973, 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701

Offset: 0

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn

Comments

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

From Fred Schneider, Sep 20 2006; edited by Danny Rorabaugh, Nov 26 2018: (Start)

Heuristic: Add the squares of several successive primes and then add successive primes until the number is abundant.

a(2) = 5^2 * 7 * 11 * 13 * 17 * 19 * 23 * 29;

a(3) = 7^2 * 11^2 * 13 * 17 * ... * 61 * 67;

a(4) = 11^2 * 13^2 * 17 * 19 * ... * 131 * 137;

a(5) = 13^2 * 17^2 * 19 * 23 * ... * 223 * 227. (End)

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

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

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 Múgica, Nov 15 2017

a(n) is the product of consecutive primes starting from prime(n+1) raised to nonincreasing powers. - Jianing Song, Apr 10 2021

From Jianing Song, Apr 14 2021: (Start)

By definition, Omega(a(n)) >= A108227(n+1) for all n, where Omega = A001222. For 0 <= n <= 12 we have Omega(a(n)) = A108227(n+1), but this is not true for n = 13, where Omega(a(13)) = 335 > A108227(14) = 334.

We also have omega(a(n)) >= A001276(n+1) for all n, where omega = A001221. The differences for known terms are 0, 0, 1, 1, 2, 3, 2, 3, 4, 4, 5, 6, 6, 6 respectively.

Conjecture: other than a(1) = 945, all terms are cubefree. (End)

Examples

			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.
		

References

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

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 0..13
• Thomas Fink, Recursively abundant and recursively perfect numbers, arXiv:2008.10398 [math.NT], 2020. Mentions this sequence.
• 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.

Crossrefs

Subsequence of A005101 and A133812; cf. A005231.

Formula

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

Extensions

2 more terms from Fred Schneider, Sep 20 2006