OFFSET
1,3
COMMENTS
Let x,y be odd numbers. Denote <+> the following binary operation: x<+>y=A000265(x+y). Let a and d be odd numbers. We call a sequence of the form b, b<+>d, (b<+>d)<+>d,... a pseudo-arithmetic progression with the initial term b and the difference d. It is not difficult to prove that every pseudo-arithmetic progression is periodic sequence. This sequence lists smallest periods of pseudo-arithmetic progressions with initial term 1 and difference 2n-1, n=1,2,...
a(n) is the number of distinct odd residues contained in set {1,2,...,2^(2*n-2)} modulo 2*n-1. Thus 2*n-1 is in A001122 iff a(n)=n-1. - Vladimir Shevelev, Jul 18 2010
LINKS
Peter J. C. Moses, Table of n, a(n) for n = 1..4096
FORMULA
EXAMPLE
For n=5, we have 1<+>9=5, 5<+>9=7, 7<+>9=1. Thus a(5)=3.
MAPLE
pseuAprog := proc(a, b) A000265(a+b) ; end proc:
A179382 := proc(n) local p, k; p := [1] ; for k from 2 do a := pseuAprog( p[-1], 2*n-1) ; if not a in p then p := [op(p), a] ; else return nops(p) ; end if; end do: end proc:
seq(A179382(n), n=1..80) ;
# R. J. Mathar, Jul 13 2010
MATHEMATICA
oddres[n_] := n/2^IntegerExponent[n, 2];
a[n_] := Module[{d = 2n-1, k=1, t=1}, While[(t = oddres[t+d])>1, k++]; k];
Array[a, 80] (* Jean-François Alcover, Apr 13 2020, translated from PARI *)
PROG
(PARI) oddres(n)=n>>valuation(n, 2)
a(n)=my(d=2*n-1, k=1, t=1); while((t=oddres(t+d))>1, k++); k
\\ Charles R Greathouse IV, May 15 2013
(Sage)
def A179382(n):
N, o, s = 2*n-1, 1, 0
while True:
o = (N + o) >> valuation(N + o, 2)
s = s + 1
if o == 1: break
return s
print([A179382(n) for n in (1..72)]) # Peter Luschny, Oct 07 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Jul 12 2010
EXTENSIONS
Corrected and extended by R. J. Mathar, Jul 13 2010
Duplicated database lines removed by R. J. Mathar, Jul 23 2010
STATUS
approved