A053006 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.

1, 4, 12, 16, 24, 25, 36, 37, 40, 45, 52, 64, 76, 81, 84, 96, 100, 109, 112, 117, 120, 132, 136, 156, 165, 169, 172, 180, 184, 192, 216, 220, 232, 240, 244, 249, 252, 256, 265, 277, 300, 301, 304, 312, 316, 324, 357, 360, 361, 364, 372, 376, 412, 420, 432

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

m is in the sequence if and only if the multiplicative order of 2 (mod 2m-1) is odd.

References

• R. K. Guy, Unsolved Problems in Number Theory, E38.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• P. Alles, On a Conjecture of J. Pelikan, J. Comb. Th. A 60 (1992) 312-313.
• N. F. J. Inglis and J. D. A. Wiseman, Very odd sequences, J. Comb. Th. A 71 (1995) 89-96.
• F. J. MacWilliams and A. M. Odlyzko, Pelikan's conjecture and cyclotomic cosets, J. Comb. Th. A 22 (1977) 110-114.

Crossrefs

Cf. A000265, A036259.

Programs

Mathematica

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 ] ] & ]
(* Second program: *)
(Select[Range[1, 999, 2], OddQ[MultiplicativeOrder[2, #]]&] + 1)/2 (* Jean-François Alcover, Dec 20 2017 *)

PARI

is(n)=znorder(Mod(2,2*n-1))%2 \\ Charles R Greathouse IV, Jun 24 2015

PARI

A000265(n)=n>>valuation(n,2)
is(n)=Mod(2,2*n-1)^A000265(eulerphi(2*n-1))==1 \\ Charles R Greathouse IV, Jun 24 2015

Python

from sympy import n_order
def A053006_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n_order(2,(n<<1)-1)&1,count(max(startvalue,1)))
A053006_list = list(islice(A053006_gen(),20)) # Chai Wah Wu, Feb 07 2023

Formula

a(n) = (A036259(n) + 1)/2.

Extensions

More terms from John W. Layman, Feb 21 2000

Additional information from Dean Hickerson, May 25 2001