[seqfan] Re: Is it certain that this is a permutation?

[hv at crypt.org]

Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
:Just submitted this:
:
:%I A175350
:%S A175350 1,2,6,5,67,11,637
:%N A175350 a(n) = the smallest positive integer not yet occurring such that the number of divisors of sum{k=1 to n} a(k) is exactly n. 
:%C A175350 It seems likely that this is a permutation of the positive integers. Is it? 
:%C A175350 sum{k=1 to n} a(k) = A175351(n). 
:%Y A175350 A175351 
:%K A175350 more,nonn
:%O A175350 1,2
:
:Is this for certain a permutation of the positive intgers? Or am I missing something obvious which proves that this is not so?
:
:(PS: I, as I bet Neil does too, hate subjective statements in the comments, such as I have here {..."seems likely"...}. As soon as this sequence appears, if any resolution has been made on this problem, then I will change the comment appropriately.)

I have no idea if this is a permutation; I suspect it will be hard to
establish.

A few more terms, if my code is accurate (a(1)..a(24)):
  1 2 6 5 67 11 637 12 348 47 57913 26 472366 463 26105 15 42488697 118
  344373650 136 2089071 2496 30991547417 7

The code below could be made rather cleverer to find more terms.

Hugo
---
#!/usr/bin/perl -w
use strict;

use Math::Pari qw(divisors isprime PARI);
my $sum = 0;
my %seen;
N: for (my $n = 1; 1; ++$n) {
	if (isprime($n)) {
		# avoid exhaustive search for prime powers
		my $power = $n - 1;
		for (my $trialsum = PARI(2); 1; ++$trialsum) {
			my $trial = $trialsum ** $power - $sum;
			next if $trial <= 0;
			try($n, $trial) and next N;
		}
	} else {
		for (my $trial = 1; 1; ++$trial) {
			try($n, $trial) and next N;
		}
	}
}
sub try {
	my($n, $trial) = @_;
	return 0 if $seen{$trial};
	my $d = divisors($sum + $trial);
	if (@$d == $n) {
		$sum += $trial;
		print "a($n) = $trial (sum $sum)\n";
		$seen{$trial} = 1;
		$d = undef;
		return 1;
	}
	$d = undef;
	return 0;
}