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

2, 5, 6, 9, 12, 13, 14, 17, 20, 21, 24, 27, 28, 29, 30, 33, 36, 37, 40, 43, 44, 45, 48, 51, 52, 55, 58, 59, 60, 61, 62, 65, 68, 69, 72, 75, 76, 77, 80, 83, 84, 87, 90, 91, 92, 93, 96, 99, 100, 103, 106, 107, 108, 111, 114, 115, 118, 121, 122, 123, 124, 125, 126, 129

Offset

1,1

Comments

Note that the lengths of the consecutive runs in a(n) form sequence A001511.

Integers that are not a sum of distinct integers of the form 2^k-1. - Vladeta Jovovic, Jan 24 2003

Also n! never ends in this many 0's in base 2 - Carl R. White, Jan 21 2008

A079559(a(n)) = 0. - Reinhard Zumkeller, Mar 18 2009

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

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

Numbers n for which A257265(m) = 0. - Reinhard Zumkeller, May 06 2015. Typo corrected by Antti Karttunen, Aug 08 2015

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Index entries for sequences related to binary expansion of n

Index entries for Colombian or self numbers and related sequences

Formula

a(n) = A080578(n+1) - 2 = A080468(n+1) + 2*n (conjectured). - Ralf Stephan, Dec 27 2013

From Antti Karttunen, Aug 08 2015: (Start)

Other identities. For all n >= 1:

A234017(a(n)) = n.

A256992(a(n)) = n.

A257126(n) = a(n) - A005187(n).

(End)

Example

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

Mathematica

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]
(* Second program: *)
t = Table[IntegerExponent[(2n)!, 2], {n, 0, 100}]; Complement[Range[t // Last], t] (* Jean-François Alcover, Nov 15 2016 *)

Prog

(Haskell)
a055938 n = a055938_list !! (n-1)
a055938_list = concat $
   zipWith (\u v -> [u+1..v-1]) a005187_list $ tail a005187_list
-- Reinhard Zumkeller, Nov 07 2011
(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
(Scheme) ;; utilizing COMPLEMENT-macro from Antti Karttunen's IntSeq-library)
(define A055938 (COMPLEMENT 1 A005187))
;; Antti Karttunen, Aug 08 2015
(Python)
def a053644(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1)
def a043545(n):
    x=bin(n)[2:]
    return int(max(x)) - int(min(x))
def a079559(n): return 1 if n==0 else a043545(n + 1)*a079559(n + 1 - a053644(n + 1))
print([n for n in range(1, 201) if a079559(n)==0]) # Indranil Ghosh, Jun 11 2017, after the comment by Reinhard Zumkeller

Crossrefs

Complement of A005187. Setwise difference of A213713 and A213717.

Row 1 of arrays A257264, A256997 and also of A255557 (when prepended with 1). Equally: column 1 of A256995 and A255555.

Cf. also arrays A254105, A254107 and permutations A233276, A233278.

Left inverses: A234017, A256992.

Cf. A001511, A046699, A079559, A080578, A086343, A227359, A227408, A234016.

Gives positions of zeros in A213714, A213723, A213724, A213731, A257265, positions of ones in A213725-A213727 and A256989, positions of nonzeros in A254110.

Cf. also A010061 (integers that are not a sum of distinct integers of the form 2^k+1).

Analogous sequence for factorial base number system: A219658, for Fibonacci number system: A219638, for base-3: A096346. Cf. also A136767-A136774.

Cf. A257508, A257509, A257126.

Sequence in context: A284657 A230506 A236072 * A190764 A276886 A047323

Adjacent sequences: A055935 A055936 A055937 * A055939 A055940 A055941

Keyword

easy,nice,nonn

Author

Alford Arnold, Jul 21 2000

Extensions

More terms from Robert G. Wilson v, Jul 24 2000

Status

approved