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A077459
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Numbers k such that k and 3*k have the same digital binary sum.
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6
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0, 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 45, 48, 51, 56, 60, 62, 63, 75, 89, 90, 93, 96, 99, 102, 103, 109, 112, 115, 120, 124, 126, 127, 139, 150, 151, 155, 177, 178, 180, 185, 186, 189, 192, 195, 198, 199, 204, 206, 207, 217, 218, 221, 224, 227, 230, 231, 237
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OFFSET
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1,2
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COMMENTS
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Numbers k such that valuation(C(3*k,k),2) = valuation(C(2*k,k),2). - Benoit Cloitre, Mar 20 2004
If n is in the sequence, then so is 2*n.
If m and n are in the sequence and 2^k>3*n then 2^k*m+n is in the sequence. (End)
Let's call "primitive" those a(n) not of the form 2^k*a(i) + a(j) with 2^k > 3*a(j), i > 1. The primitive terms not of the form 2^n-1 are (45, 89, 93, 109, 139, 151, 177, 217, 221, 237, ...) = (101101, 1011001, 1011101, 1101101, 10001011, 10010111, 10110001, 11011001, 11011101, 11101101, ...) in binary. Are all of them of the form x*2^k + 1 or 2^k + x where x = 2^m - 2^n - 1, m > n+1 (base-2 repunit with one "interior" digit 0, so that 3x has one bit 1 less than x)? - M. F. Hasler, Dec 13 2018
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LINKS
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EXAMPLE
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51 binary representation is [1, 1, 0, 0, 1, 1] and 3*51=153 binary representation is [1, 0, 0, 1, 1, 0, 0, 1], both binary sum = 4, hence 51 is in the sequence.
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MAPLE
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filter:= n -> convert(convert(n, base, 2), `+`)=convert(convert(3*n, base, 2), `+`):
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MATHEMATICA
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digitSum[n_] := Total@IntegerDigits[n, 2]; Select[Range[0, 250], digitSum[#] == digitSum[3#] &] (* Amiram Eldar, Dec 13 2018 *)
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PROG
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(Magma) [n: n in [0..300] | Valuation(Binomial(3*n, n), 2) eq Valuation(Binomial(2*n, n), 2)]; // Vincenzo Librandi, Dec 14 2018
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CROSSREFS
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KEYWORD
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base,nonn,easy
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AUTHOR
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STATUS
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approved
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