

A073122


Minimal reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n. See A072339.


6



1, 2, 5, 4, 13, 10, 9, 8, 25, 26, 29, 20, 21, 18, 17, 16, 49, 50, 53, 52, 61, 58, 57, 40, 41, 42, 45, 36, 37, 34, 33, 32, 97, 98, 101, 100, 109, 106, 105, 104, 121, 122, 125, 116, 117, 114, 113, 80, 81, 82, 85, 84, 93, 90, 89, 72, 73, 74, 77, 68, 69, 66, 65, 64, 193
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OFFSET

1,2


COMMENTS

The minimal representation is unique. The number of powers of 2 can be either even or odd. Compare with A065621, in which the number of powers of 2 is odd. The Mathematica program computes the representation for numbers 1 to 2^m. a(0) = 0.
No term has odd part congruent to 3 modulo 4.  Charlie Neder, Oct 28 2018


REFERENCES

D. E. Knuth, The Art of Computer Programming. AddisonWesley, Reading, MA, 1981, Vol. 2 (Second Edition), p. 196, (exercise 4.1. Nr. 27)


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000


FORMULA

a(2n) = a(n).
Express n as a sum of terms 2^x  2^y, x > y, such that each term defines a run of 1's in n's binary expansion. Then a(n) is the sum of all 2^x + 2^y, with the exception that a term 2^(x+1)  2^x at the end of a representation becomes 2^x.  Charlie Neder, Oct 28 2018


EXAMPLE

a(11) = 29 because 29 = 16 + 8 + 4 + 1 and 16  8 + 4  1 = 11.
a(100) = 164 because 100 in binary is 1100100. The two runs of ones correspond to 2^7  2^5 and 2^3  2^2, but since 2^3  2^2 is the last term of the representation, it can be replaced with 2^2. Therefore, a(100) = 2^7 + 2^5 + 2^2.  Charlie Neder, Oct 28 2018


MATHEMATICA

Needs["DiscreteMath`Combinatorica`"]; sumit[s_List] := Module[{i, ss=0}, Do[If[OddQ[i], ss+=s[[ i]], ss=s[[ i]]], {i, Length[s]}]; ss]; m=7; powers=Table[2^i, {i, 0, m}]; lst=Table[2m, {2^m}]; lst2=Table[0, {2^m}]; Do[t=NthSubset[i, powers]; len=Length[t]; st=sumit[t]; If[len<lst[[st]], lst[[st]]=len; lst2[[st]]=Plus@@t], {i, 2^(m+1)1}]; lst2


CROSSREFS

Cf. A065621, A072219, A072339, A256696.
Sequence in context: A121274 A256464 A111681 * A277020 A084410 A080067
Adjacent sequences: A073119 A073120 A073121 * A073123 A073124 A073125


KEYWORD

easy,nice,nonn


AUTHOR

T. D. Noe, Jul 17 2002


STATUS

approved



