A160855 - OEIS

A160855

a(n) is 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.

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

The lines that appear in the scatterplot seem to be related to the position of n in the sum of the first n 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

FORMULA

a(A236341(n)) = n. - Reinhard Zumkeller, Jul 12 2015

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