A003602 Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k.
1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42
Offset: 1
Examples
From _Peter Munn_, Jun 14 2022: (Start) Start of table showing the interleaving with the positive integers: n a(n) (n+1)/2 a(n/2) 1 1 1 2 1 1 3 2 2 4 1 1 5 3 3 6 2 2 7 4 4 8 1 1 9 5 5 10 3 3 11 6 6 12 2 2 (End)
References
- Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- N. J. A. Sloane, Table of n, a(n) for n = 1..10000
- J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
- Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle - Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.
- J.-P. Delahaye, La marelle arithmétique, Pour la Science, No. 360, October 2007. In French.
- Dale Gerdemann, Plotting Adjacent Points in A003602, Kimberling's Paraphrase, YouTube Video, 2015.
- Dale Gerdemann, Plotting Adjacent Terms of A003602 Modulo Increasing Powers of 2, YouTube Video, 2015.
- Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021.
- Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv preprint arXiv:1608.00862 [math.GM], 2016.
- Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
- Clark Kimberling, Fractal sequences
- Ralf Stephan, Some divide-and-conquer sequences ...
- Ralf Stephan, Table of generating functions
- Matty van-Son, Palindromic sequences of the Markov spectrum, arXiv:1804.10802 [math.NT], 2018.
- Eric Weisstein's World of Mathematics, Odd Part
- Index entries for sequences related to binary expansion of n
Crossrefs
Cf. A000079, A000265, A001511, A003603, A003961, A014577 (with offset 1, reduction mod 2), A025480, A035528, A048673, A101279, A110963, A117303, A126760, A181988, A220466, A249745, A253887, A337821 (2-adic valuation).
Cf. also A349134 (Dirichlet inverse), A349135 (sum with it), A349136 (Möbius transform), A349431, A349371 (inverse Möbius transform).
Cf. A264646.
Programs
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Haskell
a003602 = (`div` 2) . (+ 1) . a000265 -- Reinhard Zumkeller, Feb 16 2012, Oct 14 2010
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Haskell
import Data.List (transpose) a003602 = flip div 2 . (+ 1) . a000265 a003602_list = concat $ transpose [[1..], a003602_list] -- Reinhard Zumkeller, Aug 09 2013, May 23 2013
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Maple
A003602:=proc(n) options remember: if n mod 2 = 1 then RETURN((n+1)/2) else RETURN(procname(n/2)) fi: end proc: seq(A003602(n), n=1..83); # Pab Ter nmax := 83: for m from 0 to ceil(simplify(log[2](nmax))) do for k from 1 to ceil(nmax/(m+2)) do a((2*k-1)*2^m) := k od: od: seq(a(k), k=1..nmax); # Johannes W. Meijer, Feb 04 2013 A003602 := proc(n) a := 1; for p in ifactors(n)[2] do if op(1,p) > 2 then a := a*op(1,p)^op(2,p) ; end if; end do : (a+1)/2 ; end proc: # R. J. Mathar, May 19 2016
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Mathematica
a[n_] := Block[{m = n}, While[ EvenQ@m, m /= 2]; (m + 1)/2]; Array[a, 84] (* or *) a[1] = 1; a[n_] := a[n] = If[OddQ@n, (n + 1)/2, a[n/2]]; Array[a, 84] (* Robert G. Wilson v, May 23 2006 *) a[n_] := Ceiling[NestWhile[Floor[#/2] &, n, EvenQ]/2]; Array[a, 84] (* Birkas Gyorgy, Apr 05 2011 *) a003602 = {1}; max = 7; Do[b = {}; Do[AppendTo[b, {k, a003602[[k]]}], {k, Length[a003602]}]; a003602 = Flatten[b], {n, 2, max}]; a003602 (* L. Edson Jeffery, Nov 21 2015 *) -
PARI
A003602(n)=(n/2^valuation(n,2)+1)/2; /* Joerg Arndt, Apr 06 2011 */
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Python
import math def a(n): return (n/2**int(math.log(n - (n & n - 1), 2)) + 1)/2 # Indranil Ghosh, Apr 24 2017
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Python
def A003602(n): return (n>>(n&-n).bit_length())+1 # Chai Wah Wu, Jul 08 2022
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Scheme
(define (A003602 n) (let loop ((n n)) (if (even? n) (loop (/ n 2)) (/ (+ 1 n) 2)))) ;; Antti Karttunen, Feb 04 2015
Formula
a(n) = (A000265(n) + 1)/2.
a((2*k-1)*2^m) = k, for m >= 0 and k >= 1. - Robert G. Wilson v, May 23 2006
Inverse Weigh transform of A035528. - Christian G. Bower
G.f.: 1/x * Sum_{k>=0} x^2^k/(1-2*x^2^(k+1) + x^2^(k+2)). - Ralf Stephan, Jul 24 2003
a(2*n-1) = n and a(2*n) = a(n). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005
a(A118413(n,k)) = A002024(n,k); = a(A118416(n,k)) = A002260(n,k); a(A014480(n)) = A001511(A014480(n)). - Reinhard Zumkeller, Apr 27 2006
Ordinal transform of A001511. - Franklin T. Adams-Watters, Aug 28 2006
a(n) = A249745(A126760(A003961(n))) = A249745(A253887(A048673(n))). That is, this sequence plays the same role for the numbers in array A135764 as A126760 does for the odd numbers in array A135765. - Antti Karttunen, Feb 04 2015 & Jan 19 2016
G.f. satisfies g(x) = g(x^2) + x/(1-x^2)^2. - Robert Israel, Apr 24 2015
a(n) = A025480(n-1) + 1. - R. J. Mathar, May 19 2016
a(n) = (1 + n)/2, for n odd; a(n) = a(n/2), for n even. - David James Sycamore, Jul 28 2022
a(n) = n/2^A001511(n) + 1/2. - Alan Michael Gómez Calderón, Oct 06 2023
Extensions
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005
Comments