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

1,2

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

Links

Table of n, a(n) for n=1..49.

Wikipedia, Coefficient of variation.

Wikipedia, Standard deviation.

Example

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.

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[#]&]

Prog

(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

Crossrefs

Cf. A371383, A371384, A371463, A371464.

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

Sequence in context: A166511 A062237 A162467 * A033022 A031497 A020337

Adjacent sequences: A371459 A371460 A371461 * A371463 A371464 A371465

Keyword

nonn,base

Author

Stefano Spezia, Mar 24 2024

Status

approved