OFFSET
1,2
COMMENTS
This sequence uses the binary expansion of n rather than the decimal expansion used in the usual Keith numbers, A007629. A number n (having a t-bit binary representation) is in this sequence if n is a term in the t-step Fibonacci-like series beginning with the t bits of n. See the example below.
Let the bits of n be b(i) for i=1 to t. Then b(t+1) = Sum_{i=1..t} b(i). Subsequent terms are b(t+k+1) = 2*b(t+k) - b(k) for k=1,2,3,.... (This is equivalent to, but faster than, the usual method of adding the previous t terms to find the next term.) Due to the growth rate of the numbers in the series, the term equal to n occurs on or before position 2t in the series.
Terms in this sequence fall into families having the same number of 1 bits. For instance, 143, 285, 569, 1138, and 2276 all have 5 bits set. Numbers in each family are either 2x or 2x-1, where x is the previous number in the family. The binary expansion of each number in family f begins with f-1 (in binary).
This sequence is infinite because for any odd prime (or base-2 pseudoprime, A001567) p=2k+1, we can create a family of numbers with 2^(2k)+1 bits set. The first number in that family is 2^c + c(2^c-2)/(4^p-1) + 1, where c=2^p-1. In binary, this number is a 1 followed by a repeating pattern of p zeros and p ones and terminated by 1, for a total of 2^p bits. For example, 2212558911 is 10000011111000001111100000111111 in binary. [Typo in binary corrected by Jaroslav Krizek, Dec 09 2015]
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..30
EXAMPLE
In binary, 143 = (1,0,0,0,1,1,1,1). Subsequent terms are 5, 9, 18, 36, 72, 143.
MAPLE
isA162363 := proc(n)
local L, t, a ;
if numtheory[factorset](n) = {2} then
return false;
end if;
L := ListTools[Reverse](convert(n, base, 2)) ;
t := nops(L) ;
while true do
a := add(op(-i, L), i=1..t) ;
L := [op(L), a] ;
if a > n then
return false;
elif a = n then
return true;
end if;
end do:
end proc:
for n from 1 do
if isA162363(n) then
printf("%d, \n", n);
end if;
end do: # R. J. Mathar, Jan 12 2016
MATHEMATICA
IsKeith2[n_Integer] := Module[{b, s}, b=IntegerDigits[n, 2]; s=Total[b]; If[s<=1, n==1, k=1; While[s=2*s-b[[k]]; s<n, k++ ]; s== n]]; Select[Range[3000], IsKeith2[ # ]&]
CROSSREFS
KEYWORD
base,nonn
AUTHOR
T. D. Noe, Jul 02 2009
EXTENSIONS
Corrected name T. D. Noe, Jul 11 2009
a(22)-a(29) from Amiram Eldar, Jan 20 2026
STATUS
approved