cp's OEIS Frontend

A003076 n-th digit after decimal point of square root of n.

%I A003076 #31 Jun 22 2024 22:23:30
%S A003076 0,0,1,2,0,6,9,3,2,0,1,5,7,9,4,6,0,4,6,2,1,8,6,3,8,0,0,4,2,4,8,5,9,9,
%T A003076 0,4,0,0,3,2,1,2,4,3,9,8,9,0,1,0,4,3,4,3,5,7,0,8,4,3,7,7,5,7,0,6,2,6,
%U A003076 0,5,7,0,0,3,6,4,3,0,2,5,9,0,3,8,8,8,4,0,1,8,5,4,7,3,9,6,0,7,0,2,0,6,6,5,8
%N A003076 n-th digit after decimal point of square root of n.
%C A003076 Regarded as a decimal fraction, 0.01206932015794604621863800... is likely to be an irrational number.
%H A003076 G. C. Greubel, <a href="/A003076/b003076.txt">Table of n, a(n) for n = 0..10000</a>
%F A003076 From _M. F. Hasler_, Jun 22 2024: (Start)
%F A003076 a(n) = A000196(n*100^n) % 10, where n % 10 = A010879(n) is the final digit of n.
%F A003076 a(n) = 0 for all n in A000290 (but not only those). This explains that the value 0 is slightly more frequent than the other values. (End)
%e A003076 sqrt(0) = 0.0, so a(0) = 0 with the convention that the 0th digit after the decimal point is the digit before the decimal point,
%e A003076 sqrt(1) = 1.0, where the first digit after the decimal point is a(1) = 0,
%e A003076 sqrt(2) = 1.4142135..., where the 2nd decimal digit is a(2) = 1,
%e A003076 sqrt(3) = 1.7320508..., where the 3rd decimal digit is a(3) = 2,
%e A003076 sqrt(4) = 2.0000000..., where the 4th decimal digit is a(4) = 0,
%e A003076 sqrt(5) = 2.2360679..., where the 5th decimal digit is a(5) = 6,
%e A003076 sqrt(6) = 2.4494897..., where the 6th decimal digit is a(6) = 9, etc.
%e A003076 From _M. F. Hasler_, Jun 22 2024: (Start)
%e A003076 For the frequency of the respective digits among the first 10^k terms, we have:
%e A003076   k :   0's    1's   2's    3's   4's   5's   6's   7's   8's   9's
%e A003076 ----+---------------------------------------------------------------
%e A003076   1 :     4,     1,    2,     1,    0,    0,    1,    0,    0,    1;
%e A003076   2 :    22,     6,    9,    11,   12,    7,    8,    8,    9,    8;
%e A003076   3 :   126,   106,  105,    94,   95,   90,   86,   96,   92,  110;
%e A003076   4 :  1097,  1026, 1037,  1031,  984,  979, 1000,  956,  922,  968;
%e A003076   5 : 10320, 10053, 9926, 10122, 9855, 9985, 9934, 9857, 9855, 10093. (End)
%t A003076 Join[{0}, Array[ Function[ n, RealDigits[ N[ Sqrt[ n ], n+6 ] ]// (#[ [ 1, #[ [ 2 ] ]+n ] ])& ], 110 ]]
%t A003076 Table[ Floor[ Mod[10^n*Sqrt[n], 10]], {n, 0, 104}] (* _Robert G. Wilson v_, Jun 17 2002 *)
%o A003076 (PARI) A003076(n)=sqrtint(n*100^n)%10;
%o A003076 apply(A003076, [0..99]) \\ _M. F. Hasler_, Jun 22 2024
%Y A003076 Cf. A071901.
%K A003076 nonn,base,easy
%O A003076 0,4
%A A003076 _N. J. A. Sloane_
%E A003076 Extension and program from _Olivier Gérard_, Oct 15 1997