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A371462
Numbers such that the arithmetic mean of its digits is equal to the population standard deviation of its digits.
2
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
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
KEYWORD
nonn,base
AUTHOR
Stefano Spezia, Mar 24 2024
STATUS
approved