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

%I A179382 #32 Apr 13 2020 07:10:39
%S A179382 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,
%T A179382 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,
%U A179382 14,19,3,9,55,6,50,1,7,65,8,17,34,69,23,25,14,20,74,5,10,8,26,21
%N A179382 a(n) is the smallest period of pseudo-arithmetic progression with initial term 1 and difference 2n-1.
%C A179382 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,...
%C A179382 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
%H A179382 Peter J. C. Moses, <a href="/A179382/b179382.txt">Table of n, a(n) for n = 1..4096</a>
%F A179382 a(n) = A001222(A292239(n-1)). - _Antti Karttunen_, Oct 04 2017
%e A179382 For n=5, we have 1<+>9=5, 5<+>9=7, 7<+>9=1. Thus a(5)=3.
%p A179382 pseuAprog := proc(a,b) A000265(a+b) ; end proc:
%p A179382 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:
%p A179382 seq(A179382(n),n=1..80) ;
%p A179382 # _R. J. Mathar_, Jul 13 2010
%t A179382 oddres[n_] := n/2^IntegerExponent[n, 2];
%t A179382 a[n_] := Module[{d = 2n-1, k=1, t=1}, While[(t = oddres[t+d])>1, k++]; k];
%t A179382 Array[a, 80] (* _Jean-François Alcover_, Apr 13 2020, translated from PARI *)
%o A179382 (PARI) oddres(n)=n>>valuation(n,2)
%o A179382 a(n)=my(d=2*n-1,k=1,t=1);while((t=oddres(t+d))>1,k++);k
%o A179382 \\ _Charles R Greathouse IV_, May 15 2013
%o A179382 (Sage)
%o A179382 def A179382(n):
%o A179382     N, o, s = 2*n-1, 1, 0
%o A179382     while True:
%o A179382         o = (N + o) >> valuation(N + o, 2)
%o A179382         s = s + 1
%o A179382         if o == 1: break
%o A179382     return s
%o A179382 print([A179382(n) for n in (1..72)]) # _Peter Luschny_, Oct 07 2017
%Y A179382 Cf. A000265, A001122, A179680, A292239.
%K A179382 nonn
%O A179382 1,3
%A A179382 _Vladimir Shevelev_, Jul 12 2010
%E A179382 Corrected and extended by _R. J. Mathar_, Jul 13 2010
%E A179382 Duplicated database lines removed by _R. J. Mathar_, Jul 23 2010