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A031443
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Digitally balanced numbers: numbers which in base 2 have the same number of 0's as 1's.
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59
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2, 9, 10, 12, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197, 198, 201, 202, 204, 209, 210, 212, 216, 225, 226, 228, 232, 240
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Also numbers n such that the binary digital mean dm(2, n) = sigma(i in [1, d]: d_i * 2 - 1) / (2 * d) = 0, where d is the number of digits in the binary representation of n and d_i the individual digits. [From Reikku Kulon (reikku(AT)gmail.com), Sep 21 2008]
Contribution from Reikku Kulon (reikku(AT)gmail.com), Sep 29 2008: (Start)
Each run of values begins with 2^(2k + 1) + 2^(k + 1) - 2^k - 1. The initial values increase according to the sequence {2^(k - 1), 2^(k - 2), 2^(k - 3), ..., 2^(k - k)}.
After this, the values follow a periodic sequence of increases by successive powers of two with single odd values interspersed.
Each run ends with an odd increase followed by increases of {2^(k - k), ..., 2^(k - 2), 2^(k - 1), 2^k}, finally reaching 2^(2k + 2) - 2^(k + 1).
Similar behavior occurs in other bases. (End)
Numbers n such that A000120(n)/A031443(n) = 1/2. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Oct 15 2008]
Subsequence of A053754; A179888 is a subsequence. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 31 2010]
A000120(a(n)) = A023416(a(n)); A037861(a(n)) = 0.
A001700 gives number of terms having length 2*n in binary representation: A001700(n-1) = #{m: A070939(a(m))=2*n}. -- Reinhard Zumkeller, Jun 08 2011
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LINKS
| Reikku Kulon, Table of n, a(n) for n in [1, 10000]
Reinhard Zumkeller, Haskell Programs for Binary Digitally Balanced Numbers
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FORMULA
| a(n+1)=a(n)+2^k+2^(m-1)-1+int((2^(k+m)-2^k)/a(n))*(2^(2*m)+2^(m-1)) where k is the largest integer such that 2^k divides a(n) and m is the largest integer such that 2^m divides a(n)/2^k+1 - Ulrich Schimke (UlrSchimke(AT)aol.com)
A145037(a(n)) equals zero. [From Reikku Kulon (reikku(AT)gmail.com), Oct 02 2008]
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EXAMPLE
| 9 is present because '1001' contains 2 '0's and 2 '1's
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MAPLE
| (Maple) a:=proc(n) local nn, n1, n0: nn:=convert(n, base, 2): n1:=add(nn[i], i=1..nops(nn)): n0:=nops(nn)-n1: if n0=n1 then n else end if end proc: seq(a(n), n= 1.. 240); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 31 2008]
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MATHEMATICA
| f0[n_]:=DigitCount[n, 2, 0]; f1[n_]:=DigitCount[n, 2, 1]; f[n_]:=f1[n]/f0[n]; lst={}; Do[If[f[n]==1, AppendTo[lst, n]], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jul 21 2009]
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PROG
| (PARI) for(n=1, 100, b=binary(n); l=length(b); if(sum(i=1, l, component(b, i))==l/2, print1(n, ", ")))
(MAGMA) [ n: n in [2..250] | Multiplicity({* z: z in Intseq(n, 2) *}, 0) eq &+Intseq(n, 2) ]; // Bruno Berselli, Jun 07 2011
(Haskell) See link, showing that Ulrich Schimkes formula provides a very efficient algorithm. -- Reinhard Zumkeller, Jun 15 2011
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CROSSREFS
| Cf. A049354-A049360. See also A061854, A037861. Terms of binary width n are enumerated by A001700.
Cf. A144777, A144798, A144799, A144800, A144801, A144812 (subsets).
Cf. A000120, A001316, A006519, A145057, A145058, A145059, A145060, A144912, A145037.
Cf. A191292.
Sequence in context: A135782 A037457 A037314 * A051017 A078180 A058890
Adjacent sequences: A031440 A031441 A031442 * A031444 A031445 A031446
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KEYWORD
| nonn,base,easy,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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