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%I #31 Feb 08 2023 09:35:14
%S 1,4,12,16,24,25,36,37,40,45,52,64,76,81,84,96,100,109,112,117,120,
%T 132,136,156,165,169,172,180,184,192,216,220,232,240,244,249,252,256,
%U 265,277,300,301,304,312,316,324,357,360,361,364,372,376,412,420,432
%N Numbers m for which there exist d(1),...,d(m), each in {0,1}, such that Sum_{i=1..m-k} d(i)*d(i+k) is odd for all k=0,...,m-1.
%C m is in the sequence if and only if the multiplicative order of 2 (mod 2m-1) is odd.
%D R. K. Guy, Unsolved Problems in Number Theory, E38.
%H Amiram Eldar, <a href="/A053006/b053006.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Alles, <a href="http://dx.doi.org/10.1016/0097-3165(92)90013-K">On a Conjecture of J. Pelikan</a>, J. Comb. Th. A 60 (1992) 312-313.
%H N. F. J. Inglis and J. D. A. Wiseman, <a href="http://dx.doi.org/10.1016/0097-3165(95)90017-9">Very odd sequences</a>, J. Comb. Th. A 71 (1995) 89-96.
%H F. J. MacWilliams and A. M. Odlyzko, <a href="http://dx.doi.org/10.1016/0097-3165(77)90070-X">Pelikan's conjecture and cyclotomic cosets</a>, J. Comb. Th. A 22 (1977) 110-114.
%F a(n) = (A036259(n) + 1)/2.
%t o2[ m_ ] := Module[ {e, t}, For[ e = 1; t = 2, Mod[ t-1, m ] >0, e++, t = Mod[ 2t, m ] ]; e ]; Select[ Range[ 1, 500 ], OddQ[ o2[ 2#-1 ] ] & ]
%t (* Second program: *)
%t (Select[Range[1, 999, 2], OddQ[MultiplicativeOrder[2, #]]&] + 1)/2 (* _Jean-François Alcover_, Dec 20 2017 *)
%o (PARI) is(n)=znorder(Mod(2,2*n-1))%2 \\ _Charles R Greathouse IV_, Jun 24 2015
%o (PARI) A000265(n)=n>>valuation(n,2)
%o is(n)=Mod(2,2*n-1)^A000265(eulerphi(2*n-1))==1 \\ _Charles R Greathouse IV_, Jun 24 2015
%o (Python)
%o from sympy import n_order
%o def A053006_gen(startvalue=1): # generator of terms >= startvalue
%o return filter(lambda n:n_order(2,(n<<1)-1)&1,count(max(startvalue,1)))
%o A053006_list = list(islice(A053006_gen(),20)) # _Chai Wah Wu_, Feb 07 2023
%Y Cf. A000265, A036259.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _John W. Layman_, Feb 21 2000
%E Additional information from _Dean Hickerson_, May 25 2001
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