24384 is a member of OEIS A061561: Trajectory of 22 under the Reverse and Add! operation carried out in base 2. The terms of the sequence, up to and including 24384, are 22, 35, 84, 105, 180, 225, 360, 405, 744, 837, 1488, 1581, 3024, 3213, 6048, 6237, 12192, 12573, 24384. Even though the operations are carried out in base 2, the numbers of this sequence are shown in denary form. The actual base 2 sequence (OEIS A058042: Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2) looks like this:
10110, 100011, 1010100, 1101001, 10110100, 11100001, 101101000, 110010101, 1011101000, 1101000101, 10111010000, 11000101101, 101111010000, 110010001101, 1011110100000, 1100001011101, 10111110100000 and on and on it goes ...
22 or 10110 is chosen as the first term because it is the smallest number whose base 2 trajectory does not contain a palindrome. So starting with 10110, the reverse is 01101 and 10110 + 11101 = 100011 and so it goes.
The equivalent sequence in base 10 starts with 196 because, according to this comment for OEIS A006960, 196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.
The Reverse and Add! sequence starting with 196 looks like this:
196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007 and on and on it goes ...
ADDENDUM (added 1st June 2019):
Most numbers do become palindromes fairly quickly under the reverse and add algorithm. OEIS A023109 shows the smallest number that requires exactly \(n\) iterations of Reverse and Add to reach a palindrome. The initial terms, up to \(n=55\) and starting with \(n=0\) are:
0, 10, 19, 59, 69, 166, 79, 188, 193, 1397, 829, 167, 2069, 1797, 849, 177, 1496, 739, 1798, 10777, 6999, 1297, 869, 187, 89, 10797, 10853, 10921, 10971, 13297, 10548, 13293, 17793, 20889, 700269, 106977, 108933, 80359, 13697, 10794, 15891, 1009227, 1007619, 1009246, 1008628, 600259, 131996, 70759, 1007377, 1001699, 600279, 141996, 70269, 10677, 10833, 10911
More information at this later blog post.
10110, 100011, 1010100, 1101001, 10110100, 11100001, 101101000, 110010101, 1011101000, 1101000101, 10111010000, 11000101101, 101111010000, 110010001101, 1011110100000, 1100001011101, 10111110100000 and on and on it goes ...
22 or 10110 is chosen as the first term because it is the smallest number whose base 2 trajectory does not contain a palindrome. So starting with 10110, the reverse is 01101 and 10110 + 11101 = 100011 and so it goes.
The equivalent sequence in base 10 starts with 196 because, according to this comment for OEIS A006960, 196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.
The Reverse and Add! sequence starting with 196 looks like this:
196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007 and on and on it goes ...
ADDENDUM (added 1st June 2019):
Most numbers do become palindromes fairly quickly under the reverse and add algorithm. OEIS A023109 shows the smallest number that requires exactly \(n\) iterations of Reverse and Add to reach a palindrome. The initial terms, up to \(n=55\) and starting with \(n=0\) are:
0, 10, 19, 59, 69, 166, 79, 188, 193, 1397, 829, 167, 2069, 1797, 849, 177, 1496, 739, 1798, 10777, 6999, 1297, 869, 187, 89, 10797, 10853, 10921, 10971, 13297, 10548, 13293, 17793, 20889, 700269, 106977, 108933, 80359, 13697, 10794, 15891, 1009227, 1007619, 1009246, 1008628, 600259, 131996, 70759, 1007377, 1001699, 600279, 141996, 70269, 10677, 10833, 10911
More information at this later blog post.
No comments:
Post a Comment