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A055938 Integers not generated by b(n) = b(floor(n/2)) + n (complement of A005187).

%I A055938 #69 Mar 01 2025 12:12:05
%S A055938 2,5,6,9,12,13,14,17,20,21,24,27,28,29,30,33,36,37,40,43,44,45,48,51,
%T A055938 52,55,58,59,60,61,62,65,68,69,72,75,76,77,80,83,84,87,90,91,92,93,96,
%U A055938 99,100,103,106,107,108,111,114,115,118,121,122,123,124,125,126,129
%N A055938 Integers not generated by b(n) = b(floor(n/2)) + n (complement of A005187).
%C A055938 Note that the lengths of the consecutive runs in a(n) form sequence A001511.
%C A055938 Integers that are not a sum of distinct integers of the form 2^k-1. - _Vladeta Jovovic_, Jan 24 2003
%C A055938 Also n! never ends in this many 0's in base 2 - _Carl R. White_, Jan 21 2008
%C A055938 A079559(a(n)) = 0. - _Reinhard Zumkeller_, Mar 18 2009
%C A055938 These numbers are dead-end points when trying to apply the iterated process depicted in A071542 in reverse, i.e. these are positive integers i such that there does not exist k with A000120(i+k)=k. See also comments at A179016. - _Antti Karttunen_, Oct 26 2012
%C A055938 Conjecture: a(n)=b(n) defined as b(1)=2, for n>1, b(n+1)=b(n)+1 if n is already in the sequence, b(n+1)=b(n)+3 otherwise. If so, then see Cloitre comment in A080578. - _Ralf Stephan_, Dec 27 2013
%C A055938 Numbers n for which A257265(m) = 0. - _Reinhard Zumkeller_, May 06 2015. Typo corrected by _Antti Karttunen_, Aug 08 2015
%C A055938 Numbers which have a 2 in their skew-binary representation (cf. A169683). - _Allan C. Wechsler_, Feb 28 2025
%H A055938 T. D. Noe, <a href="/A055938/b055938.txt">Table of n, a(n) for n=1..1000</a>
%H A055938 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A055938 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F A055938 a(n) = A080578(n+1) - 2 = A080468(n+1) + 2*n (conjectured). - _Ralf Stephan_, Dec 27 2013
%F A055938 From _Antti Karttunen_, Aug 08 2015: (Start)
%F A055938 Other identities. For all n >= 1:
%F A055938 A234017(a(n)) = n.
%F A055938 A256992(a(n)) = n.
%F A055938 A257126(n) = a(n) - A005187(n).
%F A055938 (End)
%e A055938 Since A005187 begins 0 1 3 4 7 8 10 11 15 16 18 19 22 23 25 26 31... this sequence begins 2 5 6 9 12 13 14 17 20 21
%t A055938 a[0] = 0; a[1] = 1; a[n_Integer] := a[Floor[n/2]] + n; b = {}; Do[ b = Append[b, a[n]], {n, 0, 105}]; c =Table[n, {n, 0, 200}]; Complement[c, b]
%t A055938 (* Second program: *)
%t A055938 t = Table[IntegerExponent[(2n)!, 2], {n, 0, 100}]; Complement[Range[t // Last], t] (* _Jean-François Alcover_, Nov 15 2016 *)
%o A055938 (Haskell)
%o A055938 a055938 n = a055938_list !! (n-1)
%o A055938 a055938_list = concat $
%o A055938    zipWith (\u v -> [u+1..v-1]) a005187_list $ tail a005187_list
%o A055938 -- _Reinhard Zumkeller_, Nov 07 2011
%o A055938 (PARI) L=listcreate();for(n=1,1000,for(k=2*n-hammingweight(n)+1,2*n+1-hammingweight(n+1),listput(L,k)));Vec(L) \\ _Ralf Stephan_, Dec 27 2013
%o A055938 (Scheme) ;; utilizing COMPLEMENT-macro from Antti Karttunen's IntSeq-library)
%o A055938 (define A055938 (COMPLEMENT 1 A005187))
%o A055938 ;; _Antti Karttunen_, Aug 08 2015
%o A055938 (Python)
%o A055938 def a053644(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1)
%o A055938 def a043545(n):
%o A055938     x=bin(n)[2:]
%o A055938     return int(max(x)) - int(min(x))
%o A055938 def a079559(n): return 1 if n==0 else a043545(n + 1)*a079559(n + 1 - a053644(n + 1))
%o A055938 print([n for n in range(1, 201) if a079559(n)==0]) # _Indranil Ghosh_, Jun 11 2017, after the comment by _Reinhard Zumkeller_
%Y A055938 Complement of A005187. Setwise difference of A213713 and A213717.
%Y A055938 Row 1 of arrays A257264, A256997 and also of A255557 (when prepended with 1). Equally: column 1 of A256995 and A255555.
%Y A055938 Cf. also arrays A254105, A254107 and permutations A233276, A233278.
%Y A055938 Left inverses: A234017, A256992.
%Y A055938 Cf. A001511, A046699, A079559, A080578, A086343, A227359, A227408, A234016.
%Y A055938 Gives positions of zeros in A213714, A213723, A213724, A213731, A257265, positions of ones in A213725-A213727 and A256989, positions of nonzeros in A254110.
%Y A055938 Cf. also A010061 (integers that are not a sum of distinct integers of the form 2^k+1).
%Y A055938 Analogous sequence for factorial base number system: A219658, for Fibonacci number system: A219638, for base-3: A096346. Cf. also A136767-A136774.
%Y A055938 Cf. A257508, A257509, A257126.
%K A055938 easy,nice,nonn
%O A055938 1,1
%A A055938 _Alford Arnold_, Jul 21 2000
%E A055938 More terms from _Robert G. Wilson v_, Jul 24 2000