

A160855


a(n) = the smallest positive integer not occurring earlier in the sequence such that Sum_{k=1..n} a(k) written in binary contains binary n as a substring.


9



1, 3, 2, 6, 8, 4, 5, 11, 10, 24, 12, 13, 7, 9, 28, 17, 36, 14, 20, 46, 22, 44, 25, 18, 15, 16, 19, 21, 23, 26, 38, 33, 68, 30, 37, 29, 65, 39, 27, 57, 50, 88, 45, 85, 47, 83, 48, 34, 49, 51, 79, 53, 56, 32, 31, 35, 40, 41, 42, 63, 58, 72, 64, 66, 69, 61, 129, 93, 106, 60, 86
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Is this a permutation of the positive integers?
The smallest number not in {a(n)  n<=8000000} is 5083527. It appears that the quotient (a(1)+...+a(n))/n^2 meanders around between 1/2 (=perfect permutation) and 2/3: at n=8000000 the value is approximately 0.5866 (does it converge? 1/2? Golden ratio?).
The scatterplot of the first 100000 terms (see "graph") has some remarkable features which have not yet been explained.  Leroy Quet, Jul 05 2009
a(A236341(n)) = n.  Reinhard Zumkeller, Jul 12 2015
The lines that appear in the scatterplot seem to be related to the position of n in the sum of the n first terms; see colorized scatterplots in the Links section.  Rémy Sigrist, May 08 2017
From Michael De Vlieger, May 09 2017: (Start)
Starting positions of n in Sum_{k=1..n} a(k) written in binary: {1, 1, 1, 2, 1, 1, 1, 3, 2, 4, 3, 1, 1, 1, 5, 3, 2, 4, 3, 5, 4, 5, ...}.
Running total of a(n) in binary: {1, 100, 110, 1100, 10100, 11000, 11101, 101000, 110010, 1001010, 1010110, 1100011, 1101010, 1110011, ...}.
(End)


LINKS

H. v. Eitzen, Table of n, a(n) for n=1..100000
Rémy Sigrist, Colorized scatterplot of the first 100000 terms
Rémy Sigrist, Alternate colorized scatterplot of the first 100000 terms


EXAMPLE

From Michael De Vlieger, May 09 2017: (Start)
a(1) = 1 since binary n = "1" appears in the binary total of all numbers in the sequence "1" at position 1.
a(2) = 3 since binary n = "10" appears in the binary total of all numbers in the sequence (1 + 3) = "100" starting at position 1.
a(3) = 2 since binary n = "11" appears in the binary total of all numbers in the sequence (1 + 3 + 2) = "110" starting at position 1.
a(4) = 6 since binary n = "100" appears in the binary total of all numbers in the sequence (1 + 3 + 2 + 6) = "1100" starting at position 2.
...
(End)


MATHEMATICA

a = {}; Do[k = 1; While[Or[MemberQ[a, k], SequencePosition[ IntegerDigits[Total@ a + k, 2], #] == {}], k++] &@ IntegerDigits[n, 2]; AppendTo[a, k], {n, 71}]; a (* Michael De Vlieger, May 09 2017, Version 10.1 *)


PROG

(Haskell)
import Data.List (delete)
a160855 n = a160855_list !! (n  1)
a160855_list = 1 : f 2 1 [2..] where
f x sum zs = g zs where
g (y:ys) = if binSub x (sum + y)
then y : f (x + 1) (sum + y) (delete y zs) else g ys
binSub u = sub where
sub w = mod w m == u  w > u && sub (div w 2)
m = a062383 u
 Reinhard Zumkeller, Jul 12 2015


CROSSREFS

Cf. A160856.
Cf. A062383, A236341 (putative inverse).
Sequence in context: A210236 A193998 A209171 * A120232 A292961 A019444
Adjacent sequences: A160852 A160853 A160854 * A160856 A160857 A160858


KEYWORD

nonn,base,look


AUTHOR

Leroy Quet, May 28 2009


EXTENSIONS

Extended by Ray Chandler, Jun 15 2009


STATUS

approved



