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A337840 a(n) is the decimal place of the start of the first occurrence of n in the decimal expansion of n^(1/n).

%I A337840 #58 Oct 08 2020 19:21:54
%S A337840 0,4,10,1,38,6,9,4,12,17,26,0,264,144,107,101,101,4,78,68,36,86,11,17,
%T A337840 147,151,205,50,55,26,307,88,94,180,177,61,113,244,280,37,110,38,285,
%U A337840 101,124,223,243,25,86,116,66,77,146,283,3,60,20,82,27,146,82,140
%N A337840 a(n) is the decimal place of the start of the first occurrence of n in the decimal expansion of n^(1/n).
%C A337840 Does a(n) exist for all n? Some relatively large values: a(1021) = 67714, a(1111) = 64946. - _Chai Wah Wu_, Oct 07 2020
%H A337840 Chai Wah Wu, <a href="/A337840/b337840.txt">Table of n, a(n) for n = 1..10000</a>
%e A337840 For n = 1, 1^(1/1) = 1.0000000, so a(1) is 0.
%e A337840 For n = 12, 12^(1/12) ~= 1.2300755, so a(12) = 0.
%t A337840 max = 3000; a[n_] := SequencePosition[RealDigits[n^(1/n), 10, max][[1]], IntegerDigits[n]][[1, 1]] - 1; Array[a, 100] (* _Amiram Eldar_, Sep 25 2020 *)
%o A337840 (PARI) a(n) = {if (n==1, 0, my(p=10000); default(realprecision, p+1); my(x = floor(10^p*n^(1/n)), d = digits(x), nb = #Str(n)); for(k=1, #d-nb+1, my(v=vector(nb, i, d[k+i-1])); if (fromdigits(v) == n, return(k-1));); error("not found"););} \\ _Michel Marcus_, Sep 30 2020
%o A337840 (Python)
%o A337840 import gmpy2
%o A337840 from gmpy2 import mpfr, digits, root
%o A337840 gmpy2.get_context().precision=10**5
%o A337840 def A337840(n): # increase precision if -1 is returned
%o A337840     return digits(root(mpfr(n),n))[0].find(str(n)) # _Chai Wah Wu_, Oct 07 2020
%Y A337840 Cf. A177715.
%Y A337840 Decimal expansions of some n^(1/n): A002193, A002581, A005534, A011215, A011231, A011247, A011263, A011279, A011295, A011311, A011327, A011343, A011359.
%K A337840 nonn,base
%O A337840 1,2
%A A337840 _William Phoenix Marcum_, Sep 25 2020
%E A337840 More terms from _Amiram Eldar_, Sep 25 2020