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A082576
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Numbers n such that n^n has final digits the same as all digits of n.
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7
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1, 5, 6, 9, 11, 16, 21, 25, 31, 36, 41, 49, 51, 56, 57, 61, 71, 75, 76, 81, 91, 93, 96, 99, 101, 125, 151, 176, 193, 201, 249, 251, 301, 351, 375, 376, 401, 451, 499, 501, 551, 557, 576, 601, 625, 651, 693, 701, 749, 751, 776, 801, 851, 875, 901, 951, 976, 999
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OFFSET
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1,2
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COMMENTS
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n^n^n also has the same final digits as n. - Ed Pegg Jr, Jun 27 2013
For any positive integer r the sequence contains 10^r-1. - Reiner Moewald, Feb 14 2016
All terms > 96 end in 01, 25, 49, 51, 57, 75, 76, 93 or 99.
It appears that except for 1, 5, 6, 9, 57 and 93, if n is a term then so is the number obtained from n by deleting its first digit. (End)
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REFERENCES
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LINKS
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EXAMPLE
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9^9 = 387420489 ends in 9; 31^31 = 17069174130723235958610643029059314756044734431 ends in 31.
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MAPLE
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a:= proc(n) option remember; local k; for k from 1+
a(n-1) while k&^k mod (10^length(k))<>k do od; k
end: a(1):=1:
select(n -> n&^n mod 10^(1+ilog10(n)) = n, [$1..1000]); # Robert Israel, Mar 04 2016
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MATHEMATICA
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Select[Range@ 1000, Function[k, Take[IntegerDigits[#^#], -Length@ k] == k]@ IntegerDigits@ # &] (* Michael De Vlieger, Mar 04 2016 *)
Select[Range[1000], PowerMod[#, #, 10^IntegerLength[#]]==#&] (* Harvey P. Dale, Dec 21 2019 *)
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PROG
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(PARI) for (d = 1, 4, for (i = 10^(d - 1), 10^d - 1, x = Mod(i, 10^d); if (x^i == x, print(i)))) \\ David Wasserman, Oct 27 2006
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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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EXTENSIONS
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STATUS
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approved
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