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%I #64 Dec 08 2023 07:10:13
%S 1,4,7,6,5,6,3,6,9,1,1,6,3,6,5,6,7,4,9,6,1,4,7,6,5,6,3,6,9,9,1,6,3,6,
%T 5,6,7,4,9,6,1,4,7,6,5,6,3,6,9,5,1,6,3,6,5,6,7,4,9,6,1,4,7,6,5,6,3,6,
%U 9,9,1,6,3,6,5,6,7,4,9,6,1,4,7,6,5,6,3,6,9,1,1,6,3,6,5,6,7,4,9,1,1,4,7,6,5
%N Final nonzero digit of n^n.
%C The decimal number .147656369116... formed from these digits is a transcendental number; see Dresden's second article. These digits are never eventually periodic.
%C Digits appear with predictable frequencies: 1/10 for 3, 4, and 7; 1/9 for 5; 3/25 for 9; 28/225 for 1; and 307/900 for 6. - _Charles R Greathouse IV_, Oct 03 2022
%H T. D. Noe, <a href="/A116081/b116081.txt">Table of n, a(n) for n = 1..1000</a>
%H Articles can be found on <a href="http://home.wlu.edu/~dresdeng/">Dresden's Home Page</a>.
%H Gregory P. Dresden, <a href="http://www.jstor.org/stable/2691105">Two Irrational Numbers From the Last Non-Zero Digits of n! and n^n</a>, Math. Mag. 74 (October 2001), 316-320.
%H Gregory P. Dresden, <a href="http://www.jstor.org/stable/27643091">Three transcendental numbers from the last non-zero digits of n^n, F_n and n!</a>, Mathematics Magazine, pp. 96-105, vol. 81, 2008.
%H Jose María Grau and A. M. Oller-Marcen, <a href="http://arxiv.org/abs/1203.4066">On the last digit and the last non-zero digit of n^n in base b.</a>, arXiv:1203.4066 [math.NT], 2012.
%H Jose María Grau and A. M. Oller-Marcen, <a href="http://pdf.medrang.co.kr/kms01/BKMS/51/BKMS-51-5-1325-1337.pdf">On the last digit and the last non-zero digit of n^n in base b</a>, Bull. Korean Math. Soc. 51 (2014), No. 5, pp. 1325-1337.
%H S. Ikeda and K. Matsuoka, <a href="https://web.archive.org/web/20200713064910/http://siauliaims.su.lt/index.php?option=com_content&view=article&id=330&Itemid=7">On transcendental numbers generated by certain integer sequences</a>, Siauliai Math. Semin. 8 (2013), 63-69. Mentions this sequence.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F a(n) = A065881(n)^n mod 10 = A010879(A065881(n)^(A010883(n-1))). - _Robert Israel_, Oct 19 2014
%e a(4) = 6 because 4^4 (which is 256) ends in 6.
%p f:= proc(n) local d, m, p; d:= min(padic:-ordp(n,2), padic:-ordp(n,5));
%p m:= n/10^d;
%p p:= n - 1 mod 4 + 1;
%p m &^ p mod 10;
%p end proc:
%p seq(f(n), n=1..1000); # _Robert Israel_, Oct 19 2014
%t f[n_] := Block[{m = n}, While[ Mod[m, 10] == 0, m /= 10]; PowerMod[m, n, 10]]; Array[f, 105] (* _Robert G. Wilson v_, Mar 13 2006 and modified Oct 12 2014 *)
%o (PARI) f(n) = while(!(n % 10), n/=10); n % 10; \\ A065881
%o a(n) = lift(Mod(f(n), 10)^n); \\ _Michel Marcus_, Sep 13 2022
%o (PARI) a(n)=my(k=n/10^valuation(n,10)); lift(Mod(k,10)^(n%4+4)) \\ _Charles R Greathouse IV_, Sep 13 2022
%o (Python)
%o def a(n):
%o k = n
%o while k%10 == 0: k //= 10
%o return pow(k, n, 10)
%o print([a(n) for n in range(1, 106)]) # _Michael S. Branicky_, Sep 13 2022
%o (Python)
%o def A116081(n): return pow(int(str(n).rstrip('0')[-1]),n,10) # _Chai Wah Wu_, Dec 07 2023
%Y Cf. A008904, A010879, A065881.
%Y Cf. A056849.
%Y Cf. A204815, A204816, A204817, A204818, A204819, A230024, A204697.
%Y Cf. A322489, A322490.
%K easy,base,nonn,cons
%O 1,2
%A _Greg Dresden_, Mar 12 2006
%E More terms from _Robert G. Wilson v_, Mar 13 2006
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