A217445 Numbers n such that n! has the same number of terminating zeros in bases 3 and 4.

1, 2, 4, 5, 6, 7, 10, 11, 12, 13, 14, 18, 19, 21, 22, 23, 33, 36, 37, 38, 42, 43, 46, 47, 51, 56, 58, 59, 60, 61, 62, 75, 86, 88, 89, 92, 100, 101, 102, 103, 105, 112, 113, 114, 115, 120, 121, 122, 124, 125, 138, 139, 141, 147, 153, 159, 164, 166, 167, 168

Offset: 1

Author

Tanya Khovanova, Oct 03 2012

Keywords

• nonn
• base

Comments

The number of zeros of n! base 3 is approaching n/2 as n grows. Similarly, the number of zeros of n! base 4 is approaching n/2 as n grows. Consequently, this sequence is expected to have high density.

From Robert Israel, Jan 19 2017: (Start)

Numbers n such that A000120(n) + (n + A000120(n) mod 2) = A053735(n).

Since typically A000120(n) ~ log_2(n) while typically A053735(n) ~ log_3(n), the density of this sequence should go to 0, contrary to the previous comment. (End)

Comment from N. J. A. Sloane, Dec 06 2019: (Start)

Appears to be the same as the list of positive numbers n such that the last nonzero digit of n! in base 12 belongs to the set [1, 2, 5, 7, 10, 11].

The first footnote in Deshouillers et al. (2016) says: "if the last nonzero digit of n! in base 12 belongs to {1, 2, 5, 7, 10, 11} then |(digit-sum of n in base 3) - (digit-sum of n in base 2)| is <= 1; this seems to occur infinitely many times." Compare A096288. (End)

Examples

			6! is 222200 in base 3 and 23100 in base 4, both of them have 2 zeros at the end, so 6 is in the sequence.
		

References

• Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram, Bernard Landreau, Sums of the digits in bases 2 and 3, arXiv:1611.08180, 2016

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A054861 (base 3), A090616 (base 4), A090622, A096288.

Programs

Maple

s2:= n -> convert(convert(n,base,2),`+`):
s3:= n -> convert(convert(n,base,3),`+`):
select(n -> s2(n) + (n+s2(n) mod 2) = s3(n), [$1..1000]); # Robert Israel, Jan 19 2017

Mathematica

sntzQ[n_]:=Module[{f=n!},Last[Split[IntegerDigits[f,3]]]==Last[ Split[ IntegerDigits[ f,4]]]]; Select[Range[200],sntzQ] (* Harvey P. Dale, Jul 11 2020 *)

PARI

is(n)=my(L=log(n+1));sum(k=1,L\log(3),n\3^k)==sum(k=1,L\log(2),n>>k)\2 \\ Charles R Greathouse IV, Oct 04 2012

Extensions

More terms from Alois P. Heinz, Oct 03 2012