Read all lines into an array, grep the contigs, and then join them:

use strict;
use warnings;

my @data = grep { !/^(>|\s)/ } <DATA>;
chomp @data;
print join "nnnnncattccattcattaattaattaatgaatgaatgnnnnn", @data;

__DATA__
>contig number 11

tttgctcggaggggatc
>contig number 23

gaaaacacttccttattatacaggtaaaccgtatttggat
>contig number 3

aaagctcggaggggatcccct
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or a one liner using regexes:

perl -e "undef $/; print  join 'nnnnncattccattcattaattaattaatgaatgaatg
+nnnnn', <> =~ /^([tcag]+)$/gm;"  input_file.txt >output_file.txt
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