A378838 -id:A378838 - cp's OEIS frontend

A378835 Numbers with the median of the digits equal to the digital root.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 99, 109, 118, 128, 138, 148, 158, 168, 178, 181, 182, 183, 184, 185, 186, 187, 188, 190, 209, 218, 227, 237, 247, 257, 267, 272, 273, 274, 275, 276, 277, 281, 290, 309, 318, 327, 336, 346, 356, 363, 364, 365, 366, 372, 381, 390, 409, 418, 427, 436, 445

Offset: 1

Stefano Spezia, Dec 09 2024

			10 is not in the list because the median of its digits is not integer.
11 is not in the list because the median of its digits is 1 which is not equal to 2, the digital root of 11.
109 is in the list because the median of its digits is 1 which is equal to the digital root of 109.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A010888, A378836.

Cf. A378560, A378564, A378837, A378838.

A010888[n_]:=If[n==0,0,n - 9*Floor[(n-1)/9]]; Select[Range[0,450], Median[IntegerDigits[#]]==A010888[#] &]

A378836 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to the digital root.

Original entry on oeis.org

10, 1, 131, 474, 10233, 50844, 1001250, 5225775, 99980565, 536333508, 9998984322, 55188464010, 999994914558, 5683515922236, 100001648752524, 585428890525092, 10000105972653645, 60302140270087340, 1000004027662440330, 6207976859006478708, 100000111315410065850

Offset: 1

Stefano Spezia, Dec 09 2024

Chai Wah Wu, Table of n, a(n) for n = 1..43

Cf. A010888, A378835.

Cf. A378560, A378564, A378837, A378838.

A010888[n_]:=If[n==0,0,n - 9*Floor[(n-1)/9]]; a[n_]:=If[n==1,10,Module[{c=0}, For[k=10^(n-1), k<=10^n-1, k++, If[Median[IntegerDigits[k]]==A010888[k], c++]]; c]]; Array[a, 6]

from math import prod, factorial
from collections import Counter
from sympy.utilities.iterables import partitions
def A378836(n):
    if n==1: return 10
    c, f = 0, factorial(n-1)
    for i in range(1,9*n+1):
        for s,p in partitions(i,m=n,k=9,size=True):
            a = sorted(list(Counter(p).elements())+[0]*(n-s))
            b = a[len(a)-1>>1]+a[len(a)>>1]
            if b&1^1 and b>>1 == 1+(i-1)%9:
                v = list(p.values())
                p = prod((factorial(i) for i in v))*factorial(n-s)
                c += sum(f*i//p for i in v)
    return c # Chai Wah Wu, Dec 12 2024

a(11)-a(21) from Chai Wah Wu, Dec 12 2024

A378837 Numbers with the arithmetic mean of the digits equal to the digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 99, 999, 1029, 1038, 1047, 1056, 1065, 1074, 1083, 1092, 1119, 1128, 1137, 1146, 1155, 1164, 1173, 1182, 1191, 1209, 1218, 1227, 1236, 1245, 1254, 1263, 1272, 1281, 1290, 1308, 1317, 1326, 1335, 1344, 1353, 1362, 1371, 1380, 1407, 1416, 1425, 1434, 1443

Offset: 1

Stefano Spezia, Dec 09 2024

			10 is not in the list because the mean of its digits is not integer.
11 is not in the list because the mean of its digits is 1 which is not equal to 2, the digital root of 11.
1029 is in the list because the mean of its digits is 3 which is equal to the digital root of 1029.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A010888, A378838.

Cf. A378560, A378564, A378835, A378836.

A010888[n_]:=If[n==0,0,n - 9*Floor[(n-1)/9]]; Select[Range[0,1500], Mean[IntegerDigits[#]]==A010888[#] &]

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A378835 Numbers with the median of the digits equal to the digital root.

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A378836 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to the digital root.

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A378837 Numbers with the arithmetic mean of the digits equal to the digital root.

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Mathematica