A034886 -id:A034886 - cp's OEIS frontend

A332242 Numbers k such that k! has exactly k nonzero decimal digits.

1, 48, 49, 53, 54, 57

Offset: 1

Chai Wah Wu, Feb 07 2020

No other terms < 100000.

Conjecture: these 6 terms are the only terms of the sequence, i.e., there are no terms larger than 57.

			48! = 12413915592536072670862289047373375038521486354677760000000000 has 48 nonzero decimal digits, so 48 is a term.

Cf. A000142, A027869, A034886.

q:= n-> nops(subs(0=NULL, convert(n!, base, 10)))=n:
select(q, [$0..100])[];  # Alois P. Heinz, Feb 07 2020

isok(k) = #select(x->(x != 0), digits(k!)) == k; \\ Michel Marcus, Feb 08 2020

A332242_list, i, n = [], 0, 1
while i < 1000:
    s = str(n)
    if len(s) - s.count('0') == i:
        A332242_list.append(i)
    i += 1
    n *= i

{ k : A034886(k) - A027869(k) = k }.

A338973 Possible number of digits in a factorial.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 101, 102

Offset: 1

Christian Amet, Dec 18 2020

This is A034886 with duplicate terms removed.

Cf. A000142, A034886.

DeleteDuplicates @ IntegerLength[Range[71]!] (* Amiram Eldar, Dec 24 2020 *)

a(n) = A034886(n+3) for n>=4.

A348960 a(n) = floor(log_10(Pi*n!)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 96, 98, 100

Offset: 0

Paul F. Marrero Romero, Nov 05 2021

nonn

Cf. A004216, A034886, A053511.

Array[Floor@ Log10[Pi*#!] &, 71, 0] (* Michael De Vlieger, Nov 05 2021 *)

a(n) = floor(log_10(Pi*n!)).

a(n) = floor(A053511 + log_10(n!)).

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

A360181 Numbers k such that the number of odd digits in k! is greater than or equal to the number of even digits.

Original entry on oeis.org

0, 1, 11, 29, 36, 193, 281

Offset: 1

Zhining Yang, Jan 28 2023

If it exists, a(8) > 100000.

			11 is a term since 11! = 39916800, and the numbers of odd and even digits are both 4.
29 is a term since 29!=8841761993739701954543616000000, and the numbers of odd and even digits are 16 and 15 respectively.

Cf. A034886, A352547, A360182.

Select[Range[0, 500],
 Count[IntegerDigits[#!], _?OddQ] >=
   Count[IntegerDigits[#!], _?EvenQ] &]

from sympy import factorial as f
def ok(n):
    s=str(f(n))
    return(sum(1 for k in s if k in '02468')<=sum(1 for k in s if k in '13579'))
print([n for n in range(501) if ok(n)])

from math import factorial
from itertools import count, islice
def A360181_gen(startvalue=0): # generator of terms >= startvalue
    f = factorial(m:=max(startvalue,0))
    for k in count(m):
        if len(s:=str(f)) <= sum(1 for d in s if d in {'1','3','5','7','9'})<<1:
            yield k
        f *= k+1
A360181_list = list(islice(A360181_gen(),7)) # Chai Wah Wu, May 10 2023

A363743 a(n) = floor(sqrt(log_10(n!))).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Paul F. Marrero Romero, Aug 17 2023

nonn

Every nonnegative integer occurs at least 4 times.
Each integer k > 14 appears fewer than k times.
The only integers k that occur exactly k times are 11, 13 and 14.

Cf. A000196, A034886.

Mathematica
```
Array[Floor@ Sqrt[Log10[#!]] &, 93, 0]
```

a(n) = sqrtint(log(n!)/log(10)); \\ Michel Marcus, Sep 27 2023

a(n) = floor(sqrt(A034886(n) - 1)).

a(n) = A000196(A034886(n) - 1).

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A332242 Numbers k such that k! has exactly k nonzero decimal digits.

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A338973 Possible number of digits in a factorial.

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A348960 a(n) = floor(log_10(Pi*n!)).

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

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A360181 Numbers k such that the number of odd digits in k! is greater than or equal to the number of even digits.

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A363743 a(n) = floor(sqrt(log_10(n!))).

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