A061383 -id:A061383 - cp's OEIS frontend

A062181 Harmonic mean of digits is 3.

3, 26, 33, 62, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 1999, 2266, 2336, 2344, 2363, 2434, 2443, 2626, 2633, 2662, 3236, 3244, 3263, 3326, 3333, 3362, 3424, 3442, 3623, 3632, 4234, 4243, 4324, 4342, 4423, 4432, 6226, 6233, 6262, 6323, 6332

Offset: 1

Vladeta Jovovic, Jun 12 2001

Eric Weisstein's World of Mathematics, Harmonic Mean.
Wikipedia, Harmonic mean.

Cf. A062179-A062185, A061383-A061388, A061423-A061425.

Do[ h = IntegerDigits[n]; If[ Sort[h][[1]] != 0 && Length[h]/Apply[Plus, 1/h] == 3, Print[n]], {n, 1, 10^4}]

from fractions import Fraction
def hm(n):
  s = str(n)
  return None if '0' in s else len(s)/sum(Fraction(1, int(d)) for d in s)
def aupto(limit): return [m for m in range(limit+1) if hm(m) == 3]
print(aupto(6332)) # Michael S. Branicky, Mar 26 2021

More terms from Henry Bottomley, Jul 25 2001

A062184 Harmonic mean of digits is 6.

Original entry on oeis.org

6, 66, 488, 666, 848, 884, 3999, 4688, 4868, 4886, 6488, 6666, 6848, 6884, 8468, 8486, 8648, 8684, 8846, 8864, 9399, 9939, 9993, 36999, 38888, 39699, 39969, 39996, 44999, 46688, 46868, 46886, 48668, 48686, 48866, 49499, 49949, 49994, 63999

Offset: 1

Vladeta Jovovic, Jun 12 2001

Cf. A062179-A062185, A061383-A061388, A061423-A061425.

Do[ h = IntegerDigits[n]; If[ Sort[h][[1]] != 0 && Length[h]/Apply[Plus, 1/h] == 6, Print[n]], {n, 1, 10^6}]

More terms from Robert G. Wilson v, Aug 08 2001

A190296 Numbers n such that the mean of the digits in n is an integer m and n is divisible by m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 15, 20, 22, 24, 33, 40, 42, 44, 48, 51, 55, 60, 66, 77, 80, 84, 88, 99, 102, 108, 111, 114, 117, 120, 126, 132, 135, 144, 150, 153, 156, 162, 171, 180, 192, 195, 198, 201, 204, 207, 210, 216, 222, 225, 228, 234, 240, 243, 252

Offset: 1

Jaroslav Krizek, May 07 2011

Subsequence of A061383. Supersequence of A010785.

			132 is in the sequence because it is divisible by the arithmetic mean of its digits, namely 2.

okQ[n_] := Module[{m = Mean[IntegerDigits[n]]}, IntegerQ[m] && Divisible[n, m]]; Select[Range[500], okQ] (* T. D. Noe, May 09 2011 *)

Corrected and extended by T. D. Noe, May 09 2011

A350211 Numbers k such that the arithmetic mean of the digits of k! is an integer.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 12, 26, 28, 32, 59, 262, 391, 533, 579

Offset: 1

Zachary M Franco, Dec 19 2021

A heuristic argument suggests that this short list is complete. By Stirling's approximation, n! has order n*log(n) digits of which n/4 are terminal zeros. If the remaining digits are random, the mean will be just below 4.5. For n > 6, n! and also its digits sum are divisible by 9. 12! is the only factorial with 9 digits. The others have 27, 30, 36, 81, 522, 846, 1224, and 1350 digits, respectively.

			4 is a term because 4! = 24 and (2+4)/2 = 3 is an integer.

Cf. A000142, A004152, A034886, A058814, A061383.

q:= n-> (f-> (add(i, i=convert(f, base, 10))/length(f))::integer)(n!):
select(q, [$0..1000])[];  # Alois P. Heinz, Dec 19 2021

Do[If[IntegerQ[Mean[IntegerDigits[n!]]], Print[n, " ", Mean[IntegerDigits[n!]]]], {n, 1, 100000}]

isok(k) = my(d=digits(k!)); (vecsum(d) % #d) == 0; \\ Michel Marcus, Dec 19 2021

A362038 A list of lists where T(n,k) is the smallest n-digit number whose digits have arithmetic mean k, for 1 <= k <= 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 39, 59, 79, 99, 102, 105, 108, 129, 159, 189, 399, 699, 999, 1003, 1007, 1029, 1069, 1199, 1599, 1999, 5999, 9999, 10004, 10009, 10059, 10199, 10699, 12999, 17999, 49999, 99999, 100005, 100029, 100089

Offset: 1

Ctibor O. Zizka, Apr 15 2023

Subsequence of A061383. The sum of digits of T(n,k) is n*k and they are placed with the first at least 1 and the rest packed as small as possible.

			Lists begin:
         k=1     2     3     4     5     6     7     8     9
      +-----------------------------------------------------
  n=1 |    1,    2,    3,    4,    5,    6,    7,    8,    9;
  n=2 |   11,   13,   15,   17,   19,   39,   59,   79,   99;
  n=3 |  102,  105,  108,  129,  159,  189,  399,  699,  999;
  n=4 | 1003, 1007, 1029, 1069, 1199, 1599, 1999, 5999, 9999;
        ...
T(3,7) = 399 has digit mean (3+9+9)/3 = 7.

Cf. A007953, A055642, A061383.

cp's OEIS Frontend

A062181 Harmonic mean of digits is 3.

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A062184 Harmonic mean of digits is 6.

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A190296 Numbers n such that the mean of the digits in n is an integer m and n is divisible by m.

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A350211 Numbers k such that the arithmetic mean of the digits of k! is an integer.

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A362038 A list of lists where T(n,k) is the smallest n-digit number whose digits have arithmetic mean k, for 1 <= k <= 9.

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