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A228410
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The digits of a(n) and a(n+1) together can be reordered to form a palindrome; lexicographically least injective sequence of positive integers with this property.
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5
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1, 10, 100, 11, 2, 12, 21, 102, 20, 101, 22, 3, 13, 31, 103, 30, 110, 33, 4, 14, 41, 104, 40, 114, 24, 42, 112, 23, 32, 113, 34, 43, 131, 35, 5, 15, 51, 105, 50, 115, 25, 52, 121, 26, 6, 16, 61, 106, 60, 116, 36, 63, 136, 163, 316, 361, 613, 631, 1003, 111, 17, 7, 27, 72, 117, 37, 73, 137, 71, 107
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OFFSET
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1,2
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COMMENTS
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For each n=1,2,3..., choose the smallest positive integer a(n) not occurring earlier such that the digits of a(n) and the preceding term (none for n=1) taken together can form a palindrome, when suitably reordered.
This is a variant of the original version, proposed by E. Angelini, based on nonnegative integers (cf. A228407). The two sequences start with only a few terms differing and large segments in common, and one might have expected them to join a common orbit quite early, but they rather diverge more and more.
It is conjectured that the sequence is a permutation of the positive integers, i.e., that all numbers will eventually occur. To test this conjecture, one can consider the indices n at which occur the numbers equal to the smallest integer not yet used. If the conjecture is true, this is equivalent to a(m)>a(n) for all m>n; if not, then this list ends at the first missing number. These [n,a(n)] are: [1, 1], [5, 2], [12, 3], [19, 4], [35, 5], [45, 6], [62, 7], [78, 8], [88, 9], [89, 29], [92, 39], [118, 44], [149, 45], [187, 46], [314, 47], [432, 49], [477, 59], [506, 67], [507, 76], [521, 78], [531, 79], [572, 89], [573, 98], [574, 198], [954, 211][955, 222], [956, 233], [1602, 234], [1616, 235], [1623, 237], [1924, 238], [1959, 239], [2508, 258], [2515, 278], [2536, 279], [4046, 289], [4047, 298], [4053, 489], [4054, 498], ...
Sequence A228412 is an "arithmetic" variant, where instead of the union of the digits, the sum of terms is considered. Sequence A062932 is a further variant where injectivity is replaced by monotonicity.
Sequences A231433 and A231442 are variants where "palindrome" is replaced with "prime".
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LINKS
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PROG
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(PARI) {u=0; a=1; for(n=1, 99, u+=1<<a; print1(a", "); for(k=1, 9e9, bittest(u, k)&&next; d=vecsort(Vec(Str(a, k)), , 4); d[2]=="0"&&next; s=!bittest(#d, 0); forstep(i=2, #d, 2, d[i-1]==d[i]&&next; s&&next(2); s=d[i--]); a=k; break))}
(Python)
from collections import Counter
A228410_list, l, s, b = [1], Counter('1'), 2, set()
for _ in range(10**2):
....i = s
....while True:
........if i not in b:
............li, o = Counter(str(i)), 0
............for d in (l+li).values():
................if d % 2:
....................if o > 0:
........................break
....................o += 1
............else:
................A228410_list.append(i)
................l = li
................b.add(i)
................while s in b:
....................b.remove(s)
....................s += 1
................break
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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