A371462 - OEIS

%I #21 Mar 31 2024 04:39:20

%S 0,10,20,30,40,50,60,70,80,90,1001,1010,1014,1041,1049,1094,1100,1104,

%T 1140,1401,1409,1410,1490,1904,1940,2002,2020,2028,2082,2200,2208,

%U 2280,2802,2820,3003,3030,3300,4004,4011,4019,4040,4091,4101,4109,4110,4190,4400,4901,4910

%N Numbers such that the arithmetic mean of its digits is equal to 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.

%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 1014 is a term since the mean of the digits is (1 + 0 + 1 + 4)/4 = 3/2 and the standard deviation of the digits is sqrt(((1-3/2)^2 + (0-3/2)^2 + (1-3/2)^2 + (4-3/2)^2)/4) = sqrt((1/4 + 9/4 + 1/4 + 25/4)/4) = sqrt(9/4) = 3/2.

%p filter:= proc(x) local F,n,mu,i;

%p   F:= convert(x,base,10);

%p   n:= nops(F);

%p   mu:= convert(F,`+`)/n;

%p   evalb(2*mu^2 = add(F[i]^2,i=1..n)/n)

%p end proc:

%p select(filter, [$0..10000]); # _Robert Israel_, Mar 24 2024

%t DigStd[n_]:=If[n==0||IntegerLength[n]==1, 0, Sqrt[(IntegerLength[n]-1)/IntegerLength[n]]StandardDeviation[IntegerDigits[n]]]; Select[Range[0, 5000], Mean[IntegerDigits[#]]==DigStd[#]&]

%o (Python)

%o from itertools import count, islice

%o def A371462_gen(startvalue=0): # generator of terms >= startvalue

%o     return filter(lambda n:sum(map(int,(s:=str(n))))**2<<1 == len(s)*sum(int(d)**2 for d in s), count(max(startvalue,0)))

%o A371462_list = list(islice(A371462_gen(),20)) # _Chai Wah Wu_, Mar 28 2024

%Y Cf. A371383, A371384, A371463, A371464.

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

%K nonn,base

%O 1,2

%A _Stefano Spezia_, Mar 24 2024