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A230517 An irrational x such that the decimal representation of neither x nor sqrt(x) contains the digit 0.

%I A230517 #19 Oct 20 2019 01:59:42
%S A230517 1,2,1,3,2,1,1,3,2,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,2,1,3,1,1,1,1,1,
%T A230517 2,1,1,3,1,3,1,1,1,2,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,3,1,1,2,1,1,1,
%U A230517 1,3,1,3,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1
%N A230517 An irrational x such that the decimal representation of neither x nor sqrt(x) contains the digit 0.
%C A230517 The rational number 1/9 is an example of a number in [0, 1] such that the decimal representation of neither x nor sqrt(x) contains the digit 0. The object of Problem 10439 of the Amer. Math. Monthly was to find an irrational with the same property (see link).
%C A230517 The solution proposed by _Jerrold Grossman_ defines a sequence of irrationals starting with c1= 0.121121112... (A042974). Moving from left to right, the 0's in the decimal expansion of sqrt(cn) are eliminated by increasing the corresponding digit in the decimal expansion of cn by 2. The limit of cn is a number with the desired property.
%C A230517 The indices of the decimals that are successively changed are 4, 8, 29, 38, 40, 54, 62, 70, 72, 96, 118, ... (see print(ndeci) in PARI script).
%C A230517 The decimal expansion of sqrt(x) begins with 0.3483118317127931144162557719319698175373163374567....
%H A230517 C. V. Eynden, <a href="http://www.jstor.org/stable/2975298">Problem 10439. An irrational mimic of 1/9</a>, Amer. Math. Monthly, 104 (1997), 873.
%e A230517 0.12132113211112111112111111213111112113131112111111111411111113...
%o A230517 (PARI) pdeci(x, nb) = {x = x * 10; for (n=1, nb, d = floor(x); x = (x-d)*10; print1(d, ", ");); print();}
%o A230517 finddeci(x) = {x = x * 10; found = 0; nd = 1; while (! found, d = floor(x); x = (x-d)*10; if (d == 0, found = 1, nd++);); nd;}
%o A230517 changedeci(x, ndeci) = {deci = floor(x * 10^ndeci) - 10*floor(x * 10^(ndeci-1)); x += 2/10^ndeci; x;}
%o A230517 lista(nn) = {prec = 2*nn; default(realprecision, prec); x = 0; for (n=1, prec, x = 10*x + 1 + issquare(9+8*n);); x /= 10^prec; ok = 0; while (! ok, y = sqrt(x); ndeci = finddeci(y); print1(ndeci, ", "); x = changedeci(x, ndeci); if (ndeci > nn, ok =1);); print(); pdeci(x, nn); print("sqrt(x)=", sqrt(x));} \\ _Michel Marcus_, Oct 22 2013
%Y A230517 Cf. A042974, A042975.
%K A230517 nonn,base,cons
%O A230517 0,2
%A A230517 _Michel Marcus_, Oct 22 2013