Abundant numbers

[David Wilson]

How about the smallest abundant number not divisible by the first n primes (n >= 
0).
----- Original Message ----- 
From: <hv at crypt.org>
To: <seqfan at ext.jussieu.fr>
Cc: <franktaw at netscape.net>
Sent: Thursday, February 09, 2006 7:35 AM
Subject: Re: Abundant numbers


> Earlier I wrote:
> :franktaw at netscape.net wrote:
> ::Based on the recent discussion of "First even term", I have been inspired
> ::to submit the following sequence:
> ::
> ::%S A000001 
> 12,945,20,945,12,5391411025,12,945,20,81081,12,5391411025,12,6435,56,945,12,
> ::5391411025,12,81081,20,945,12,5391411025,12,945,20,6435,12,
> ::20169691981106018776756331,12,945,20,945,12,5391411025,12,945,20,81081,12,
> ::366005822969340125,12,945,56,945,12,5391411025,12,81081
> ::%N A000001 Smallest abundant number relatively prime to n.
> :[...]
> ::I would appreciate it if someone would check these before I actually submit
> ::the sequence.
> :
> :I can confirm the figures; the code below takes me 22s to calculate
> :up to a(210) = 49061132957714428902152118459264865645885092682687973.
>
> It is also interesting to consider "a(n) relatively prime to n" as a
> special case of "a(n) fixes certain prime powers".
>
> a(n) is uninteresting for squareful n: a(xy^2) = a(xy) for all x, y.
> If b(n), n = \prod{ p_i^x_i } is defined as the least abundant
> k = \prod{ p_i^y_i } such that y_i = x_i - 1 whenever x_i > 0,
> we get a new sequence identical to a(n) above for squarefree n,
> but with the additional property that every abundant k appears
> in the sequence at least at a(k * \prod{ p | k }) = k.
>
> I'll submit this once Franklin's sequence is in, so I can correct the
> crossrefs (or Franklin can submit the two together if that is convenient).
>
> It may also be interesting to consider the sequence of abundant numbers
> that appear earlier than their guaranteed spot in b(n).
>
> %I A000002
> %S A000002 12,945,20,18,12,5391411025,12,12,12,81081,12,70,12,6435,56,24,12
> %T A000002 5775,12,18,20,945,12,20,20,945,18,18,12,20169691981106018776756331
> %U A000002 12,48,20,945,12,30,12,945,20,12,12,366005822969340125,12,18,12,945
> %N A000002 Smallest abundant number with some prime powers fixed by n.
> %C A000002 If n = \prod{ p_i^x_i }, a(n) = \prod{ p_i^y_i } then we require 
> y_i = x_i - 1 whenever x_i > 0.
> %C A000002 Same as A000001 for squarefree n.
> %C A000002 If k is abundant then k = a(k * \prod{ p | k })
> %e A000002 12 = 2^2.3^1, so a(12)=70 is the least abundant 2^1.3^0.k with 
> (k,2.3)=1
> %Y A000002 Cf A005101, A005231, A047802, A000001.
> %O A000002 1,1
> %K A000002 nonn
> %A A000002 Hugo van der Sanden (hv at crypt.org), Feb 9 2006
>
> Hugo