Search: a257509 -id:a257509
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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
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OFFSET
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1,1
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COMMENTS
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It seems that for all n >= 0, Sum_{k=1 .. 2^n} a(k) = 2^(n+3).
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LINKS
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FORMULA
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PROG
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(Haskell)
a256489 n = a256489_list !! (n-1)
a256489_list = zipWith (-) (tail a257509_list) a257509_list
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A055938
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Integers not generated by b(n) = b(floor(n/2)) + n (cf. A005187).
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+10
84
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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
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OFFSET
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1,1
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COMMENTS
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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
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
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LINKS
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FORMULA
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Other identities. For all n >= 1:
(End)
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a055938 n = a055938_list !! (n-1)
a055938_list = concat $
zipWith (\u v -> [u+1..v-1]) a005187_list $ tail a005187_list
(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)
(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))
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CROSSREFS
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Cf. also A010061 (integers that are not a sum of distinct integers of the form 2^k+1).
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KEYWORD
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easy,nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A257265
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Distance to n from a leaf nearest to n in the binary beanstalk.
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+10
8
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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
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OFFSET
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0,1
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COMMENTS
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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).
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LINKS
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FORMULA
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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.]
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EXAMPLE
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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.
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PROG
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(Scheme)
;; 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))))))))))
(Haskell)
a257265 = length . us where
us n = if a079559 n == 0
then [] else () : zipWith (const $ const ())
(us $ a213723 n) (us $ a213724 n)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A257508
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Next-to-leaf vertices in binary beanstalk; Numbers n for which A257265(n) = 1.
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+10
7
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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
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OFFSET
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1,2
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COMMENTS
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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.
Equal to A257507 with duplicate terms removed.
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LINKS
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EXAMPLE
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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.
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PROG
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(Haskell)
a257508 n = a257508_list !! (n-1)
a257508_list = filter ((== 1) . a257265) [0..]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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