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

hv at crypt.org hv at crypt.org
Tue Apr 20 11:50:44 CEST 2010

```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;
}

```