A049355 Digitally balanced numbers in base 4: equal numbers of 0's, 1's, ... 3's.

75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, 210, 216, 225, 228, 16815, 16827, 16830, 16875, 16878, 16890, 17007, 17019, 17022, 17055, 17079, 17085, 17115, 17118, 17127, 17133, 17142, 17145, 17259, 17262, 17274, 17307, 17310, 17319, 17325

Offset: 1

Author

Harvey P. Dale

Keywords

• nonn
• base

Comments

The sum of reciprocals, Sum_{n>=1} 1/a(n), converges. In general, the sum of the reciprocals of balanced numbers in base b converges for all b >= 4, and diverges for b = 2 or 3 (Papanicolaou, 2013). Grivaux (2015) gives 3 * Sum_{k>=1} (4*k)!/(k!^4 * 4^(4*k)) = 0.857... as an upper bound for this sum. The sum is converging slowly: the sums of the reciprocals of the terms with no more than 4*k digits in base 4, for k = 1, 2, ..., are 0.129.., 0.183..., 0.213..., 0.233..., 0.248..., 0.260..., 0.269..., 0.276..., 0.282..., 0.288..., ... . - Amiram Eldar, Feb 15 2024

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1001
• Vassilis Papanicolaou, Problem 11729, The American Mathematical Monthly, Vol. 120, No. 8 (2013), p. 754; Summing the Reciprocals of Normal Numbers Base b, Solution to Problem 11729 by Nicole Grivaux, ibid., Vol. 122, No. 8 (2015), p. 806.
• Roberto Tauraso, Problem 11729.

Crossrefs

Cf. A031443.

Row n = 4 of A378000.

Programs

Mathematica

Select[Range[20000],Length[Union[DigitCount[#,4]]]==1&] (* Harvey P. Dale, Mar 19 2013 *)
FromDigits[#,4]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},4n,{1,0,2,3}],{n,2}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)

PARI

is(n) = {my(c = matreduce(digits(n,4))[,2]); #c == 4 && #Set(c) == 1;} \\ Amiram Eldar, Feb 15 2024

Extensions

Offset corrected by Amiram Eldar, Feb 15 2024