A213715 -id:A213715 - cp's OEIS frontend

A213716 Complement of A213715.

6, 10, 12, 14, 18, 20, 22, 25, 27, 29, 30, 34, 36, 38, 41, 43, 45, 46, 49, 51, 53, 55, 57, 59, 61, 62, 66, 68, 70, 73, 75, 77, 78, 81, 83, 85, 87, 89, 91, 93, 94, 97, 99, 101, 103, 105, 106, 108, 109, 111, 113, 114, 116, 117, 119, 121, 122, 124, 125, 126, 130, 132, 134, 137, 139, 141, 142, 145, 147, 149, 151, 153, 155, 157, 158, 161, 163, 165, 167, 169, 170, 172, 173, 175, 177, 178, 180, 181, 183, 185

Offset: 1

Antti Karttunen, Oct 26 2012

nonn

Used to compute A213717. Such i, that A005187(i) is not in A179016.

Antti Karttunen, Table of n, a(n) for n = 1..10000

A218541 First differences of A213715.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

This sequence seems to share similar fractal characteristics with A213712. See the given link to graph-plot.

Antti Karttunen, Table of n, a(n) for n = 0..8727
OEIS Server, Sequence plotted together with A213712 showing similar fractal features.

(define (A218541 n) (- (A213715 (1+ n)) (A213715 n)))

a(n) = A213715(n+1)-A213715(n).

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

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

A213714 Inverse function for injection A005187.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 0, 4, 5, 0, 6, 7, 0, 0, 0, 8, 9, 0, 10, 11, 0, 0, 12, 13, 0, 14, 15, 0, 0, 0, 0, 16, 17, 0, 18, 19, 0, 0, 20, 21, 0, 22, 23, 0, 0, 0, 24, 25, 0, 26, 27, 0, 0, 28, 29, 0, 30, 31, 0, 0, 0, 0, 0, 32, 33, 0, 34, 35, 0, 0, 36, 37, 0, 38, 39, 0, 0, 0, 40, 41, 0, 42, 43, 0, 0, 44, 45, 0, 46, 47, 0

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

a(0)=0; thereafter if n occurs as a term of A005187, a(n)=its position in A005187, otherwise zero. This works as an "inverse" function for A005187 in a sense that a(A005187(n)) = n for all n.

a(n)*A234017(n) = 0 for all n.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Can be used when computing A213715, A213723, A213724, A233275, A233277. Cf. A005187, A046699, A079559, A234017, A230414.

import Data.List (genericIndex)
a213714 n = genericIndex a213714_list n
a213714_list = f [0..] a005187_list 0 where
   f (x:xs) ys'@(y:ys) i | x == y    = i : f xs ys (i+1)
                         | otherwise = 0 : f xs ys' i
-- Reinhard Zumkeller, May 01 2015

from sympy import factorial
def a046699(n):
    if n<3: return 1
    s=1
    while factorial(2*s)%(2**(n - 1))>0: s+=1
    return s
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))
def a(n): return 0 if n==0 else a079559(n)*(a046699(n + 2) - 1) # Indranil Ghosh, Jun 11 2017

a(0)=0, for n>0, a(n) = A079559(n) * (A046699(n+2)-1) [With A046699's October 2012 starting offset. Incorrect indexing shown in this formula corrected by Antti Karttunen, Dec 18 2013]

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A213716 Complement of A213715.

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A218541 First differences of A213715.

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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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A213714 Inverse function for injection A005187.

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