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A049355
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Digitally balanced numbers in base 4: equal numbers of 0's, 1's, ... 3's.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) is(n) = {my(c = matreduce(digits(n, 4))[, 2]); #c == 4 && #Set(c) == 1; } \\ Amiram Eldar, Feb 15 2024
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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