A213717 -id:A213717 - cp's OEIS frontend

A213731 a(n)=0 if n is in A055938, a(n)=1 if n is in A179016, otherwise (i.e., n is in A213717), a(n)=2.

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

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

Those natural numbers n for which a(n)=1 belong to the infinite trunk of "Carl White's beanstalk" (see A179016), while the numbers n for which a(n)=0, are the leaves (terminal, dead-end nodes) of the same beanstalk, while those n for which a(n)=2, are non-terminal nodes in its finite tendrils.

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A179016, A213725, A213726, A213730.

: (define (A213731 n) (+ (A079559 n) (- (A079559 n) (A213719 n))))

a(n) = 2*A079559(n) - A213719(n).

A213722 a(n) = number of terms in A213717 whose magnitude is in range [(2^n)-1,(2^(n+1))-2] (or equally, in range [(2^n),(2^(n+1))-1]).

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 15, 34, 74, 158, 333, 694, 1434, 2946, 6030, 12312, 25090, 51040, 103654, 210165, 425488, 860267, 1737263, 3504655, 7063602, 14225185, 28627843, 57579313

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

a(n) = A011782(n)-A213709(n).

a(n) = Sum_{i=(2^n) .. (2^(n+1))-1} (A079559(i)-A213719(i)).

A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 11, 15, 16, 19, 23, 26, 31, 32, 35, 39, 42, 46, 49, 53, 57, 63, 64, 67, 71, 74, 78, 81, 85, 89, 94, 97, 101, 104, 109, 112, 116, 120, 127, 128, 131, 135, 138, 142, 145, 149, 153, 158, 161, 165, 168, 173, 176, 180, 184, 190, 193, 197, 200, 205, 209

Offset: 0

Carl R. White, Jun 24 2010

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "binary beanstalk" from its root (zero). The name "beanstalk" is due to Antti Karttunen.

There are many finite sequences such as 0,1,2; 0,1,3,4,7,9; etc. obeying the same condition (see A218254) and as the length increases, so (necessarily) does the similarity to this infinite sequence.

Alois P. Heinz and Antti Karttunen, Table of n, a(n) for n = 0..16405 (first 1000 terms from Alois P. Heinz)
Paul Tek, Illustration of the first terms

Cf. A000120, A010062, A011371, A213710, A213711, A213717, A213730, A213731, A218600, A218616, A218789, A233271, A218602, A054429. First differences: A213712, complement: A213713.
A subsequence of A005187, i.e., a(n) = A005187(A213715(n)). For all n,
A071542(a(n)) = n, and furthermore A213708(n) <= a(n) <= A173601(n). (Cf. A218603, A218604).
Rows of A218254, when reversed, converge towards this sequence.
Cf. A276623, A219648, A219666, A255056, A276573, A276583, A276613 for analogous constructions, and also A259934.

TakeWhile[Reverse@ NestWhileList[# - DigitCount[#, 2, 1] &, 10^3, # > 0 &], # <= 209 &] (* Michael De Vlieger, Sep 12 2016 *)

a(0)=0, a(1)=1, and for n > 1, if n = A218600(A213711(n)) then a(n) = (2^A213711(n)) - 1, and in other cases, a(n) = a(n+1) - A213712(n+1). (This formula is based on Carl White's observation that this iterated/converging path must pass through each (2^n)-1. However, it would be very interesting to know whether the sequence admits more traditional recurrence(s), referring to previous, not to further terms in the sequence in their definition!) - Antti Karttunen, Oct 26 2012
a(n) = A218616(A218602(n)). - Antti Karttunen, Mar 04 2013
a(n) = A054429(A233271(A218602(n))). - Antti Karttunen, Dec 12 2013

Starting offset changed from 1 to 0 by Antti Karttunen, Nov 05 2012

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

A213725 a(n)=0 if n is in the infinite trunk of Carl White's beanstalk (i.e., in A179016), otherwise 1 + number of steps to reach the farthest leaf in that finite branch of the beanstalk.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2

Offset: 0

Antti Karttunen, Nov 01 2012

Antti Karttunen, Table of n, a(n) for n = 0..16384

Differs from A213726 for the first time at n=208, where a(n)=5, while A213726(208)=6.

For all n, a(A179016(n)) = 0, a(A055938(n)) = 1, and a(A213717(n)) >= 2. Cf. A213726, A213727-A213731.

If A079559(n)=0, a(n)=1; otherwise, if A213719(n)=1, a(n)=0; otherwise a(n) = 1+max(a(A213723(n)), a(A213724(n))),

A213727 a(n) = 0 if n is in the infinite trunk of the "binary beanstalk", otherwise number of nodes (including leaves and the node n itself) in that finite branch of the beanstalk.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 3, 0, 1, 1, 1, 0, 0, 1, 3, 0, 1, 1, 5, 0, 1, 3, 0, 1, 1, 1, 1, 0, 0, 1, 3, 0, 1, 1, 5, 0, 1, 3, 0, 1, 1, 1, 0, 7, 1, 0, 5, 1, 1, 0, 3, 1, 3, 0, 1, 1, 1, 1, 1, 0, 0, 1, 3, 0, 1, 1, 5, 0, 1, 3, 0, 1, 1, 1, 0, 7, 1, 0, 5, 1, 1, 0, 3

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

a(n) tells, for each natural number n, whether it belongs to the infinite trunk of the binary beanstalk (in A179016, when a(n)=0), or if it is one of the terminal nodes (i.e., leaves, A055938, when a(n)=1), or otherwise, when a(n) > 1, tells from how many different numbers one can arrive at this n by repeatedly subtracting their bit-count (A000120) from them (as explained in A071542), and including also n itself as one of the cases.

a(n) is always odd unless it is zero. In particular, each a(A213717(n)) is an odd number greater than one.

			a(10) = 3 because we include 10 itself ("1010" in binary) and the two numbers n for which it is true that n - A000120(n) = 10, i.e., 12 and 13 ("1100" and "1101" in binary). Furthermore, there do not exist any such numbers for 12 or 13, as both are members of A055938 (see also the comment at A213717).
Similarly, a(22) = 5 as there are the following five cases: 22 itself, 24 as 24-A000120(24) = 24-2 = 22 (note that 24 is in A055938), 25 as 25-A000120(25) = 25-3 = 22, and the two terminal nodes (leaves) branching from 25, that is, 28 & 29 (as 28-A000120(28) = 28-3 = 25, and 29-A000120(29) = 29-4=25).

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A179016.
Cf. also A055938, A079559, A213717, A213719, A213723, A213724, A213725, A213726, A213731.
Analogous sequences computed for other "beanstalk" or similar tree systems: A227643, A230427, A255327.

If A079559(n)=0, a(n)=1; otherwise, if A213719(n)=1, a(n)=0; otherwise a(n) = 1+a(A213723(n))+a(A213724(n)).
Other identities. For all n:
a(A179016(n)) = 0, a(A055938(n)) = 1, and a(A213717(n)) >= 3.
a(A213717(n)) = (2*A213726(A213717(n)))-1.

A213713 Complement of A179016.

Original entry on oeis.org

2, 5, 6, 9, 10, 12, 13, 14, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 61, 62, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 84, 86, 87, 88, 90, 91, 92, 93, 95, 96, 98, 99, 100, 102, 103, 105, 106, 107, 108, 110, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 124, 125, 126, 129

Offset: 1

Antti Karttunen, Oct 26 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 1..15000

Union of A055938 & A213717.

A213726 a(n)=0 if n is in the infinite trunk of the "beanstalk" (i.e., in A179016), otherwise number of terminal nodes (leaves) in that finite branch of the beanstalk.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 0, 3, 1, 1, 1, 0, 2, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 0, 3, 1, 1, 1, 0, 2, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 0, 3, 1, 1, 1, 0, 2, 1, 6

Offset: 0

Antti Karttunen, Nov 01 2012

a(n) tells for each natural number n, whether it belongs to the infinite trunk of beanstalk (when a(n)=0), or if it is one of the terminal nodes (i.e., leaves, A055938) (when a(n)=1), or otherwise, when a(n)>1, tells from how many terminal nodes one can end to this n, by repeatedly subtracting their bit count (A000120) from them (as explained in A071542).

			a(10)=2 because the only numbers in A055938 from which one can end to 10 by the process described in A071542/A179016 are 12 and 13 (see comment at A213717). Similarly, a(22)=3 as there are following three cases: 24 as 24-A000120(24) = 24-2 = 22, and also 28 & 29 as 28-A000120(28) = 28-3 = 25, and 29-A000120(29) = 29-4 = 25, and then 25-A000120(25) = 25-3 = 22.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Differs from A213725 for the first time at n=208, where a(n)=6, while A213725(208)=5.

For all n, a(A179016(n)) = 0, a(A055938(n)) = 1, and a(A213717(n)) >= 2. For all n, A213727(A213717(n)) = (2*a(A213717(n)))-1. Cf. A213725-A213731. Records: A218548, A218549.

If A079559(n)=0, a(n)=1; otherwise, if A213719(n)=1, a(n)=0; otherwise a(n) = a(A213723(n))+a(A213724(n)).

A213715 a(n) = position of A179016(n) in A005187.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 26, 28, 31, 32, 33, 35, 37, 39, 40, 42, 44, 47, 48, 50, 52, 54, 56, 58, 60, 63, 64, 65, 67, 69, 71, 72, 74, 76, 79, 80, 82, 84, 86, 88, 90, 92, 95, 96, 98, 100, 102, 104, 107, 110, 112, 115, 118, 120, 123, 127, 128, 129, 131, 133, 135, 136, 138, 140, 143, 144, 146, 148, 150, 152, 154, 156, 159, 160, 162, 164, 166, 168

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8727

Cf. A213716, A213717.

: (define (A213715 n) (A213714 (A179016 n)))

a(n) = A213714(A179016(n)).

Offset changed from 1 to 0 by Antti Karttunen, Nov 05 2012

A257512 Those vertices of the binary beanstalk whose children are both leaves.

Original entry on oeis.org

10, 18, 25, 34, 41, 54, 56, 66, 73, 86, 88, 102, 110, 117, 119, 130, 137, 150, 152, 166, 174, 181, 183, 198, 206, 213, 222, 229, 243, 244, 246, 258, 265, 278, 280, 294, 302, 309, 311, 326, 334, 341, 350, 357, 371, 372, 374, 390, 398, 405, 414, 421, 435, 436, 446, 453, 467, 468, 483, 491, 498, 499, 501, 514

Offset: 1

Antti Karttunen, May 03 2015

nonn

Numbers n for which both A079559(A213723(n)) and A079559(A213724(n)) are zero.

Numbers which occur twice in A257507.

			10 is present, because A011371(12) = A011371(13) = 10, and both 12 and 13 are terms of A055938. See also Paul Tek's illustration.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Paul Tek, Illustration of how natural numbers in range 0 .. 133 are organized as a binary tree in the binary beanstalk

Cf. A011371, A055938, A079559, A213723, A213724, A257507, A257265.
First differences: A256490.
Subsequence of A005187, A213717 and A257508.

cp's OEIS Frontend

A213731 a(n)=0 if n is in A055938, a(n)=1 if n is in A179016, otherwise (i.e., n is in A213717), a(n)=2.

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A213722 a(n) = number of terms in A213717 whose magnitude is in range [(2^n)-1,(2^(n+1))-2] (or equally, in range [(2^n),(2^(n+1))-1]).

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A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

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Mathematica

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

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A213725 a(n)=0 if n is in the infinite trunk of Carl White's beanstalk (i.e., in A179016), otherwise 1 + number of steps to reach the farthest leaf in that finite branch of the beanstalk.

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A213727 a(n) = 0 if n is in the infinite trunk of the "binary beanstalk", otherwise number of nodes (including leaves and the node n itself) in that finite branch of the beanstalk.

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A213713 Complement of A179016.

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A213726 a(n)=0 if n is in the infinite trunk of the "beanstalk" (i.e., in A179016), otherwise number of terminal nodes (leaves) in that finite branch of the beanstalk.

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A213715 a(n) = position of A179016(n) in A005187.

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