A062289 Numbers n such that n-th row in Pascal triangle contains an even number, i.e., A048967(n) > 0.

2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001

Keywords

• nonn

Comments

Numbers n such that binary representation contains the bit string "10". Union of A043569 and A101082. - Rick L. Shepherd, Nov 29 2004

The asymptotic density of this sequence is 1 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10001
• Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016.
• Oliver Kullmann and Xishun Zhao, On variables with few occurrences in conjunctive normal forms, in: K. A. Sakallah and L. Simon (eds), International Conference on Theory and Applications of Satisfiability Testing, Springer, Berlin, Heidelberg, 2011, pp. 33-46; arXiv preprint, arXiv:1010.5756 [cs.DM], 2010-2011.
• Oliver Kullmann and Xishun Zhao, Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses, arXiv preprint, arXiv:1505.02318 [cs.DM], 2015.

Crossrefs

Complement of A000225, so these might be called non-Mersenne numbers.

A132782 is a subsequence.

Cf. A048967, A057716, A043569, A101082.

Cf. A007461, A036987, A102370, A105027.

Cf. A261461, A261922.

Programs

Haskell

a062289 n = a062289_list !! (n-1)
a062289_list = 2 : g 2 where
   g n = nM n : g (n+1)
   nM k = maximum $ map (\i -> i + min i (a062289 $ k-i+1)) [2..k]
   -- Cf. link [Oliver Kullmann, Xishun Zhao], Def. 3.1, page 3.
-- Reinhard Zumkeller, Feb 21 2012, Dec 31 2010

Mathematica

ok[n_] := MatchQ[ IntegerDigits[n, 2], {_, 1, 0, _}]; Select[ Range[100], ok] (* Jean-François Alcover, Dec 12 2011, after Rick L. Shepherd *)

PARI

isok(m) = #select(x->((x%2)==0), vector(m+1, k, binomial(m, k-1))); \\ Michel Marcus, Jan 26 2021

Python

def A062289(n): return n+(m:=n.bit_length())-(not n>=(1<Chai Wah Wu, Jun 30 2024

Formula

a(n) = A057716(n+1) - 1.

a(n) = 2 if n=1, otherwise max{min{2*i, a(n-i+1) + i}: 1 < i <= n}.

A036987(a(n)) = 0. - Reinhard Zumkeller, Mar 06 2012

A007461(a(n)) mod 2 = 0. - Reinhard Zumkeller, Apr 02 2012

A102370(n) = A105027(a(n)). - Reinhard Zumkeller, Jul 21 2012

A261461(a(n)) = A261922(a(n)). - Reinhard Zumkeller, Sep 17 2015

Extensions

More terms from Rick L. Shepherd, Nov 29 2004