A371462 Numbers such that the arithmetic mean of its digits is equal to the population standard deviation of its digits.

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 1001, 1010, 1014, 1041, 1049, 1094, 1100, 1104, 1140, 1401, 1409, 1410, 1490, 1904, 1940, 2002, 2020, 2028, 2082, 2200, 2208, 2280, 2802, 2820, 3003, 3030, 3300, 4004, 4011, 4019, 4040, 4091, 4101, 4109, 4110, 4190, 4400, 4901, 4910

Offset: 1

Author

Stefano Spezia, Mar 24 2024

Keywords

• nonn
• base

Comments

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

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

The concatenation of several copies of any term is a term. - Robert Israel, Mar 24 2024

Examples

			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.
		

Links

• Wikipedia, Coefficient of variation.
• Wikipedia, Standard deviation.

Crossrefs

Cf. A371383, A371384, A371463, A371464.

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

Programs

Maple

filter:= proc(x) local F,n,mu,i;
  F:= convert(x,base,10);
  n:= nops(F);
  mu:= convert(F,`+`)/n;
  evalb(2*mu^2 = add(F[i]^2,i=1..n)/n)
end proc:
select(filter, [$0..10000]); # Robert Israel, Mar 24 2024

Mathematica

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[#]&]

Python

from itertools import count, islice
def A371462_gen(startvalue=0): # generator of terms >= startvalue
    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)))
A371462_list = list(islice(A371462_gen(),20)) # Chai Wah Wu, Mar 28 2024