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A307048 Permutation of the positive integers derived from the terms of A322469 having the form 6*k - 2.

%I A307048 #12 Nov 13 2024 19:05:38
%S A307048 2,1,6,5,10,4,14,7,18,13,22,8,26,3,30,21,34,12,38,9,42,29,46,16,50,23,
%T A307048 54,37,58,20,62,19,66,45,70,24,74,17,78,53,82,28,86,39,90,61,94,32,98,
%U A307048 15,102,69,106,36,110,25,114,77,118,40,122,55
%N A307048 Permutation of the positive integers derived from the terms of A322469 having the form 6*k - 2.
%C A307048 The sequence is the flattened form of an irregular table U(i, j) similar to table T(i, j) in A322469. U(i, j) = k is defined only for the elements T(i, j) which have the form 6*k - 2, so the table is sparsely filled.
%C A307048 Like in A322469, the columns in table U contain arithmetic progressions.
%C A307048 a(n) is a permutation of the positive integers, since A322469 is one, and since there is a one-to-one mapping between any a(n) = k and some A322469(m) = 6*k - 2.
%C A307048 There is a hierarchy of such permutations of the positive integers derived by mapping the terms of the form 6*k - 2 to k:
%C A307048   Level 1: A322469
%C A307048   Level 2: A307048 (this sequence)
%C A307048   Level 3: A160016 = 2, 1, 4, 6, 8, 3, ... period of (3 even, 1 odd number)
%C A307048   Level 4: A000027 = 1, 2, 3, 4 ... (the positive integers)
%C A307048   Level 5: A000027
%e A307048 Table U(i, j) begins:
%e A307048    i\j   1  2  3  4  5  6  7
%e A307048    -------------------------
%e A307048    1:
%e A307048    4:          2
%e A307048    7:                   1
%e A307048   10:
%e A307048   13:          6
%e A307048   16:                5
%e A307048   19:
%e A307048   22:         10
%e A307048   25:             4
%e A307048   28:
%e A307048   31:         14
%e A307048 -----
%e A307048 T(4, 3) = 10 = 6*2 - 2, therefore U(4, 3) = 2.
%e A307048 T(7, 6) =  4 = 6*1 - 2, therefore U(7, 6) = 1.
%o A307048 (Perl)
%o A307048 # Derived from A322469
%o A307048 use integer; my $n = 1; my $i = 1; my $an;
%o A307048 while ($i <= 1000) { # next row
%o A307048   $an = 4 * $i - 1; &term();
%o A307048   while ($an % 3 == 0) {
%o A307048     $an /= 3; &term();
%o A307048     $an *= 2; &term();
%o A307048   } # while divisible by 3
%o A307048   $i ++;
%o A307048 } # while next row
%o A307048 sub term {
%o A307048   if (($an + 2) % 6 == 0) {
%o A307048     my $bn = ($an + 2) / 6;
%o A307048     print "$n $bn\n"; $n ++;
%o A307048   }
%o A307048 }
%Y A307048 Cf. A000027, A160016, A322469.
%K A307048 nonn,easy
%O A307048 1,1
%A A307048 _Georg Fischer_, Mar 21 2019