A179382 a(n) is the smallest period of pseudo-arithmetic progression with initial term 1 and difference 2n-1.

1, 1, 2, 1, 3, 5, 6, 1, 4, 9, 2, 4, 10, 9, 14, 1, 5, 5, 18, 4, 10, 7, 5, 9, 10, 2, 26, 8, 9, 29, 30, 1, 6, 33, 11, 14, 3, 9, 15, 17, 27, 41, 2, 11, 4, 4, 3, 14, 24, 15, 50, 23, 4, 53, 18, 14, 14, 19, 3, 9, 55, 6, 50, 1, 7, 65, 8, 17, 34, 69, 23, 25, 14, 20, 74, 5, 10, 8, 26, 21

Offset: 1

Author

Vladimir Shevelev, Jul 12 2010

Keywords

• nonn

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

Examples

			For n=5, we have 1<+>9=5, 5<+>9=7, 7<+>9=1. Thus a(5)=3.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..4096

Crossrefs

Cf. A000265, A001122, A179680, A292239.

Programs

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 *)

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

Formula

a(n) = A001222(A292239(n-1)). - Antti Karttunen, Oct 04 2017

Extensions

Corrected and extended by R. J. Mathar, Jul 13 2010

Duplicated database lines removed by R. J. Mathar, Jul 23 2010