A257509 -id:A257509 - cp's OEIS frontend

A256489 First differences of A257509: a(n) = A257509(n+1) - A257509(n).

8, 8, 3, 13, 3, 7, 11, 11, 3, 7, 11, 4, 12, 8, 3, 16, 3, 7, 11, 4, 12, 8, 3, 8, 13, 8, 3, 13, 3, 7, 9, 16, 3, 7, 11, 4, 12, 8, 3, 8, 13, 8, 3, 13, 3, 7, 9, 7, 14, 8, 3, 13, 3, 7, 9, 13, 3, 7, 9, 6, 10, 7, 5, 20, 3, 7, 11, 4, 12, 8, 3, 8, 13, 8, 3, 13, 3, 7, 9, 7, 14, 8, 3, 13, 3, 7, 9, 13, 3, 7, 9, 6, 10, 7, 5, 10, 15, 8, 3, 13, 3, 7, 9

Offset: 1

Antti Karttunen, May 03 2015

It seems that for all n >= 0, Sum_{k=1 .. 2^n} a(k) = 2^(n+3).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A257509, A256490.

a256489 n = a256489_list !! (n-1)
a256489_list = zipWith (-) (tail a257509_list) a257509_list
-- Reinhard Zumkeller, May 06 2015

(define (A256489 n) (- (A257509 (+ n 1)) (A257509 n)))

a(n) = A257509(n+1) - A257509(n).

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

Original entry on oeis.org

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

Alford Arnold, Jul 21 2000

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
Numbers which have a 2 in their skew-binary representation (cf. A169683). - Allan C. Wechsler, Feb 28 2025

			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

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.

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

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 *)

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

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

;; utilizing COMPLEMENT-macro from Antti Karttunen's IntSeq-library)
(define A055938 (COMPLEMENT 1 A005187))
;; Antti Karttunen, Aug 08 2015

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)

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

A257265 Distance to n from a leaf nearest to n in the binary beanstalk.

Original entry on oeis.org

2, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2

Offset: 0

Antti Karttunen, Apr 29 2015

nonn

If n is one of the terms of A055938 (that is, one of the leaves of the binary beanstalk), then a(n) is zero. Otherwise, a(n) is the least number of iterations of A011371 required to reach n, when we start from any term of A055938.
Compare to the definition of A213725, which gives 1 + distance to the farthest leaf in the binary beanstalk (giving 0 for the nodes which reside in A179016, the infinite stem of the beanstalk, as there the maximum distance would not have a finite value).
Note that although the recursive formula given here mirrors the one given at A213725, it cannot be implemented in a naive way, precisely because of that infinite stem, as it would lead to recursion without end. Instead, with ordinary eagerly evaluating programming languages we have to employ, for example, a breadth-first search, as in the given Scheme-program.
Question: Will there ever appear a term larger than 2? (Only terms 0 - 2 occur in range 0 .. 2097151).

			For 0, the nearest leaf is 2, as when we start from 2, and always subtract the binary weight, A000120, we have: 2 - A000120(2) = A011371(2) = 1, and A011371(1) = 0, thus it takes two steps to get to 0, and there are no other terms of A055938 from which it would take fewer steps), so a(0) = 2, and also a(1) = 1, because it's one step nearer to 2.
a(2) = 0, because 2 is one of the terms of A055938.
a(8) = 2, because 12, 13 and 14 are the three nearest leaves to 8, and A011371(12) = A011371(13) = 10, A011371(14) = 11, A011371(10) = A011371(11) = 8 (thus it takes two iterations of A011371 to reach 8 from any of those three leaves) and there are no leaves nearer.
Please see also Paul Tek's illustration.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Ela, peano
Haskell Wiki, Peano numbers
Paul Tek, Illustration of how natural numbers in range 0 .. 133 are organized as a binary tree in the binary beanstalk

Cf. A055938 (positions of 0's), A257508 (of 1's), A257509 (of 2's).
Cf. also A000120, A011371, A079559, A213723, A213724.
Cf. also A179016, A213725, A257264.

a257265 = length . us where
   us n = if a079559 n == 0
             then [] else () : zipWith (const $ const ())
                               (us $ a213723 n) (us $ a213724 n)
-- Reinhard Zumkeller, May 03 2015

;; Do a breadth-first search over the descendants, which are at each step of iteration sorted by their distance from the starting node.
(define (A257265 n) (let loop ((descendants (list (cons 0 n)))) (let ((dist (caar descendants)) (node (cdar descendants))) (cond ((zero? (A079559 node)) dist) (else (loop (sort (append (list (cons (+ 1 dist) (A213724 node)) (cons (+ 1 dist) (A213723 node))) (cdr descendants)) (lambda (a b) (< (car a) (car b))))))))))

If A079559(n) = 0, then a(n) = 0, otherwise a(n) = 1 + min(a(A213723(n)), a(A213724(n))). [But please see the comments above.]

A257508 Next-to-leaf vertices in binary beanstalk; Numbers n for which A257265(n) = 1.

Original entry on oeis.org

1, 3, 4, 7, 10, 11, 15, 18, 22, 23, 25, 26, 31, 34, 38, 39, 41, 46, 47, 49, 50, 54, 56, 57, 63, 66, 70, 71, 73, 78, 79, 81, 82, 86, 88, 94, 95, 97, 98, 102, 104, 105, 110, 113, 116, 117, 119, 120, 127, 130, 134, 135, 137, 142, 143, 145, 146, 150, 152, 158, 159, 161, 162, 166, 168, 169, 174, 177, 180, 181

Offset: 1

Antti Karttunen, May 03 2015

nonn

Numbers n for which A257265(n) = 1, in other words, numbers n for which a descendant leaf nearest to n in binary beanstalk is one edge away.
Numbers n such that either A079559(A213723(n)) or A079559(A213724(n)) (or both) are zero.
Equal to A257507 with duplicate terms removed.

			3 is present because it has an immediate leaf-child 5, as A011371(5) = 3.
4 is present because it has an immediate leaf-child 6, as A011371(6) = 4.
10 is present because it has two immediate leaf-children, 12 and 13, as A011371(12) = A011371(13) = 10.
See also Paul Tek's illustration.

Antti Karttunen, Table of n, a(n) for n = 1..12289
Paul Tek, Illustration of how natural numbers in range 0 .. 133 are organized as a binary tree in the binary beanstalk

Positions of 1's in A257265.
Subsequence of A005187.
Cf. A011371, A079559, A213723, A213724, A257507, A257509, A257512 (a subsequence).

a257508 n = a257508_list !! (n-1)
a257508_list = filter ((== 1) . a257265) [0..]
-- Reinhard Zumkeller, May 06 2015

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A256489 First differences of A257509: a(n) = A257509(n+1) - A257509(n).

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

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A257265 Distance to n from a leaf nearest to n in the binary beanstalk.

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A257508 Next-to-leaf vertices in binary beanstalk; Numbers n for which A257265(n) = 1.

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