A249609 - OEIS

%I #22 Jul 13 2024 16:37:16

%S 0,0,0,1,2,1,1,0,0,1,1,3,1,2,7,1,2,1,1,3,1,2,2,1,1,2,2,1,2,1,1,4,5,1,

%T 1,3,1,3,2,1,1,3,4,1,2,1,1,6,1,2,6,1,2,1,1,3,3,1,1,2,1,2,6,1,2,1,1,3,

%U 1,3,2,1,1,2,2,1,9,1,1,2,1,4,2,1,2,1,1

%N a(n) is the smallest m, 1<=m<=n, such that binomial(n,m) is evil (A001969); a(n)=0 if there is no such m.

%C Conjecture: there are only five n: 0,1,2,7,8, for which all entries of the n-th Pascal row (A007318) are odious (A000069). _Peter J. C. Moses_ verified the conjecture up to n = 10^6.

%C Positions of records are 0,3,4,11,14,76,...; see A249650.

%C Conjecture verified up to n = 5*10^10. - _Michael S. Branicky_, Jul 13 2024

%H Peter J. C. Moses, <a href="/A249609/b249609.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 1, iff n is evil.

%t evilQ:=EvenQ[First[DigitCount[#,2]]]&;

%t Table[If[#>n,0,#]&[NestWhile[#+1&,1,!evilQ[Binomial[n,#]]&]],{n,0,100}] (* _Peter J. C. Moses_, Nov 03 2014 *)

%o (Python)

%o from math import comb

%o from itertools import count

%o def A249609(n):

%o     for m in range(1, n+1):

%o         if comb(n, m).bit_count()&1 == 0: return m

%o     return 0

%o print([A249609(n) for n in range(87)]) # _Michael S. Branicky_, Jul 13 2024

%Y Cf. A000069, A001969, A007318, A249650.

%K nonn,base

%O 0,5

%A _Vladimir Shevelev_, Nov 02 2014

%E More terms from _Peter J. C. Moses_, Nov 02 2014