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A166133
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After initial 1,2,4, a(n+1) is the smallest divisor of a(n)^2-1 that has not yet appeared in the sequence.
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26
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1, 2, 4, 3, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13, 21, 20, 19, 18, 17, 24, 23, 22, 69, 28, 27, 26, 25, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 201, 80, 79
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OFFSET
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1,2
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COMMENTS
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The initial 1,2,4 provides the smallest example with this rule that is not simply the integers in order, nor (apparently) ends with all divisors of a(n)^2-1 already present.
Apparently the sequence is infinite and includes every positive integer.
The sequence contains many runs of incrementing and decrementing values. In the 1200 steps following the 4, there are 136 increments, 706 decrements, and 358 larger steps. What is the limiting distribution for these steps? [Click the "listen" button to appreciate these runs. - N. J. A. Sloane, Apr 03 2015]
After 3, 198, 270, 570, 522, 600, 822, and 882, we have a(n+1) = a(n)^2-1. Does this happen infinitely often? Cf. A256406, A256407.
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LINKS
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EXAMPLE
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After a(24) = 22, the divisors of 22^2-1 = 483 are 1, 3, 7, 21, 23, 69, 161, and 483; 1, 3, 7, 21, and 23 have already occurred, so a(25) = 69.
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MATHEMATICA
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s = {1, 2, 4}; e = 4; Do[d = Divisors[e^2 - 1]; i = 1;
While[MemberQ[s, d[[i]]], i++]; e = d[[i]]; AppendTo[s, e], {19997}]; s (* Hans Havermann, Apr 03 2015 *)
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PROG
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(PARI) al(n, m=4, u=6)={local(ds, db);
u=bitor(u, 1<<m); print1(m);
for(i=1, n,
ds=divisors(m^2-1);
for(k=2, #ds, m=ds[k]; db=1<<m; if(!bitand(u, db), break));
u=bitor(u, db); print1(", "m))}
/* This prints the sequence without the initial 1, 2. */
(Haskell)
import Data.List (delete); import Data.List.Ordered (isect)
a166133 n = a166133_list !! (n-1)
a166133_list = 1 : 2 : 4 : f (3:[5..]) 4 where
f zs x = y : f (delete y zs) y where
y = head $ isect (a027750_row' (x ^ 2 - 1)) zs
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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