A371383 -id:A371383 - cp's OEIS frontend

A371384 a(n) is the denominator of the arithmetic mean of the digits of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

Stefano Spezia, Mar 20 2024

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences.

Cf. A004426, A004427, A007953, A055642, A061383, A371383 (numerator).

a[n_]:=Denominator[Mean[IntegerDigits[n]]]; Array[a,90,0]

from math import gcd
def A371384(n): return (l:=len(s:=str(n)))//gcd(l,sum(map(int,s))) # Chai Wah Wu, Mar 22 2024

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

Original entry on oeis.org

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

Stefano Spezia, Mar 24 2024

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

			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.

Wikipedia, Coefficient of variation.
Wikipedia, Standard deviation.

Cf. A371383, A371384, A371463, A371464.

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

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

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[#]&]

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

A371463 Numbers such that the arithmetic mean of its digits is equal to twice the population standard deviation of its digits.

Original entry on oeis.org

0, 13, 26, 31, 39, 62, 93, 1133, 1313, 1331, 1779, 1797, 1977, 2266, 2626, 2662, 3113, 3131, 3311, 3399, 3939, 3993, 6226, 6262, 6622, 7179, 7197, 7719, 7791, 7917, 7971, 9177, 9339, 9393, 9717, 9771, 9933, 10111, 11011, 11101, 11110, 11123, 11132, 11213, 11231

Offset: 1

Stefano Spezia, Mar 24 2024

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

			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.

Wikipedia, Coefficient of variation.
Wikipedia, Standard deviation.

Cf. A371383, A371384, A371462, A371464.

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

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[#]&]

from itertools import count, islice
def A371463_gen(startvalue=0): # generator of terms >= startvalue
    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)))
A371463_list = list(islice(A371463_gen(),20)) # Chai Wah Wu, Mar 30 2024

A371464 Numbers such that the arithmetic mean of its digits is equal to three times the population standard deviation of its digits.

Original entry on oeis.org

0, 12, 21, 24, 36, 42, 48, 63, 84, 1122, 1212, 1221, 2112, 2121, 2211, 2244, 2424, 2442, 2556, 2565, 2655, 3366, 3447, 3474, 3636, 3663, 3744, 4224, 4242, 4347, 4374, 4422, 4437, 4473, 4488, 4734, 4743, 4848, 4884, 5256, 5265, 5526, 5562, 5625, 5652, 6255, 6336, 6363

Offset: 1

Stefano Spezia, Mar 24 2024

Equivalently, numbers whose digits have the coefficient of variation (or relative population standard deviation) equal to 1/3.
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

			2244 is a term since the mean of the digits is (2 + 2 + 4 + 4)/4 = 3 and the standard deviation of the digits is sqrt(((2-3)^2 + (2-3)^2 + (4-3)^2 + (4-3)^2)/4) = 1.

Wikipedia, Coefficient of variation.
Wikipedia, Standard deviation.

Cf. A371383, A371384, A371462, A371463.

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

DigStd[n_]:=If[n==0||IntegerLength[n]==1, 0, Sqrt[(IntegerLength[n]-1)/IntegerLength[n]]StandardDeviation[IntegerDigits[n]]]; Select[Range[0, 6400], Mean[IntegerDigits[#]]==3DigStd[#]&]

from itertools import count, islice
def A371464_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:10*sum(s:=tuple(map(int,str(n))))**2 == 9*len(s)*sum(d**2 for d in s), count(max(startvalue,0)))
A371464_list = list(islice(A371464_gen(),20)) # Chai Wah Wu, Mar 30 2024

A383305 a(n) is number of n-digit nonnegative integers whose difference between the largest and smallest digits is equal to the arithmetic mean of its digits.

Original entry on oeis.org

1, 6, 39, 266, 1730, 11361, 74809, 494194, 3273132, 21730506, 144588345, 964050593, 6440655572, 43111601819, 289112380019, 1942335481170, 13072051432742, 88125501965430, 595077180675348, 4024698113281006, 27261843502415806, 184931926767687963, 1256249015578188517, 8545135121520262849, 58198759816476208605

Offset: 1

Stefano Spezia, Apr 22 2025

Cf. A037904, A054054, A054055, A371383, A371384, A383304.

a[n_]:=Module[{c=KroneckerDelta[n,1]}, For[k=10^(n-1), k<=10^n, k++, If[Max[d=IntegerDigits[k]]-Min[d]==Mean[d], c++]]; c]; Array[a,7]

def A383305(n):
    if n<=1: return n
    s={(k,k,k):1 for k in range(1,10)}
    for i in range(n-1):
        snew={}
        for (h,l,t),v in s.items():
            for d in range(10):
                p=(max(h,d),min(l,d),t+d)
                if p in snew:
                    snew[p]+=v
                else:
                    snew[p]=v
        s=snew
    return sum( v for (h,l,t),v in s.items() if n*(h-l)==t) # Bert Dobbelaere, Apr 25 2025

More terms from Bert Dobbelaere, Apr 25 2025

A383304 Nonnegative integers whose difference between the largest and smallest digits is equal to the arithmetic mean of its digits.

Original entry on oeis.org

0, 13, 26, 31, 39, 62, 93, 123, 132, 144, 213, 225, 231, 246, 252, 264, 267, 276, 288, 312, 321, 348, 369, 384, 396, 414, 426, 438, 441, 462, 483, 522, 624, 627, 639, 642, 672, 693, 726, 762, 828, 834, 843, 882, 936, 963, 1133, 1223, 1232, 1313, 1322, 1331, 1344, 1434, 1443

Offset: 1

Stefano Spezia, Apr 22 2025

			144 is a term since 4 - 1 = 3 = (1 + 4 + 4)/3.

Cf. A037904, A054054, A054055, A179239, A371383, A371384, A383305.

Subsequence of A061383.

Select[Range[0,1500], Max[d=IntegerDigits[#]]-Min[d]==Mean[d] &]

def ok(n): return sum(d:=list(map(int, str(n)))) == (max(d) - min(d))*len(d)
print([k for k in range(1500) if ok(k)]) # Michael S. Branicky, Apr 23 2025

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A371384 a(n) is the denominator of the arithmetic mean of the digits of n.

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

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A371463 Numbers such that the arithmetic mean of its digits is equal to twice the population standard deviation of its digits.

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A371464 Numbers such that the arithmetic mean of its digits is equal to three times the population standard deviation of its digits.

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A383305 a(n) is number of n-digit nonnegative integers whose difference between the largest and smallest digits is equal to the arithmetic mean of its digits.

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A383304 Nonnegative integers whose difference between the largest and smallest digits is equal to the arithmetic mean of its digits.

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Python