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a(n) = smallest k such that the base 4 Reverse and Add! trajectory of A075421(n) joins the trajectory of k.
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266, 3, 719, 795, 799, 269, 258, 286, 4207, 1037, 4236, 4278, 256, 4169, 4182, 4189, 271, 4338, 4402, 4598, 4662, 4108, 312, 5357, 6157, 4104, 4159, 7247, 7295, 7407, 7549, 8063, 4157, 8189, 4141, 12431, 12463, 12539, 15487, 4349, 4239, 7391, 16522
COMMENTS
a(n) determines a 1-1-mapping from the terms of A075421 to the terms of A091675. For the inverse mapping cf. A091677.
EXAMPLE
A075421(1) = 290, the trajectory of 290 ( A075299) joins the trajectory of 266 = A091675(12) at 4195, so a(1) = 266. A075421(6) = 1210, the trajectory of 1210 joins the trajectory of 269 = A091675(13) at 17975, so a(6) = 269.
a(n) = smallest non-palindromic k such that the base-2 Reverse and Add! trajectory of k is palindrome-free and joins the trajectory of A092210(n).
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2
26, 65649, 89, 4193, 3599, 775, 68076, 2173
COMMENTS
Terms a(9) to a(29) are 205796147 (conjectured), 4402, 16720, 1089448, 442, 537, unknown, 1050177, 1575, 28822, unknown, 40573, 1066, 1587, unknown, unknown, 1081, 1082, 1085, 1115, 4185.
a(n) determines a 1-to-1 mapping from the terms of A092210 to the terms of A075252, the inverse of the mapping determined by A092211. The 1-to-1 property of the mapping depends on the conjecture that the base-2 Reverse and Add! trajectory of each term of A092210 contains only a finite number of palindromes (cf. A092215).
EXAMPLE
A092210(3) = 64, the trajectory of 64 joins the trajectory of 89 at 48480, so a(3) = 89. A092210(5) = 98, the trajectory of 98 joins the trajectory of 3599 = A075252(16) at 401104704, so a(5) = 3599.
MATHEMATICA
limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = NestList[# + IntegerReverse[#, 2] &, 1, limit];
A092210 = Flatten@{1, Select[Range[2, 266], (l =
Length@NestWhileList[# + IntegerReverse[#, 2] &, #, !
MemberQ[utraj, #] &, 1, limit];
utraj =
Union[utraj, NestList[# + IntegerReverse[#, 2] &, #, limit]];
l == limit + 1) &]};
For[i = 1, i <= Length@ A092210, i++, itraj = NestList[# + IntegerReverse[#, 2] &, A092210[[i]], limit]; While[ktraj =
NestWhileList[# + IntegerReverse[#, 2] &,
k, # != IntegerReverse[#, 2] &, 1, limit];
PalindromeQ[k] || Length@ktraj != limit + 1 || ! IntersectingQ[itraj, ktraj], k++];
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