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Numbers such that the arithmetic mean of its digits is equal to twice the population standard deviation of its digits.
2

%I #19 Mar 30 2024 15:56:46

%S 0,13,26,31,39,62,93,1133,1313,1331,1779,1797,1977,2266,2626,2662,

%T 3113,3131,3311,3399,3939,3993,6226,6262,6622,7179,7197,7719,7791,

%U 7917,7971,9177,9339,9393,9717,9771,9933,10111,11011,11101,11110,11123,11132,11213,11231

%N Numbers such that the arithmetic mean of its digits is equal to twice the population standard deviation of its digits.

%C Equivalently, numbers whose digits have the coefficient of variation (or relative population standard deviation) equal to 1/2.

%C Any number obtained without leading zeros from a permutation of the digits of a given term of the sequence is also a term.

%C The concatenation of several copies of any term is a term. - _Robert Israel_, Mar 24 2024

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Coefficient_of_variation">Coefficient of variation</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Standard_deviation">Standard deviation</a>.

%e 1133 is a term since the mean of the digits is (1 + 1 + 3 + 3)/4 = 2 and the standard deviation of the digits is sqrt(((1-2)^2 + (1-2)^2 + (3-2)^2 + (3-2)^2)/4) = 1.

%t DigStd[n_]:=If[n==0||IntegerLength[n]==1, 0, Sqrt[(IntegerLength[n]-1)/IntegerLength[n]]StandardDeviation[IntegerDigits[n]]]; Select[Range[0, 12000], Mean[IntegerDigits[#]]==2DigStd[#]&]

%o (Python)

%o from itertools import count, islice

%o def A371463_gen(startvalue=0): # generator of terms >= startvalue

%o return filter(lambda n:5*sum(s:=tuple(map(int,str(n))))**2 == len(s)*sum(d**2 for d in s)<<2, count(max(startvalue,0)))

%o A371463_list = list(islice(A371463_gen(),20)) # _Chai Wah Wu_, Mar 30 2024

%Y Cf. A371383, A371384, A371462, A371464.

%Y Cf. A238619, A238620, A238658, A238660, A238662.

%K nonn,base

%O 1,2

%A _Stefano Spezia_, Mar 24 2024