[seqfan] New conjectures for odious and evil numbers

Vladimir Shevelev shevelev at bgu.ac.il
Sun Sep 25 17:54:59 CEST 2011

Dear SeqFans,

Consider a system of residues modulo 4: R={0,1,2,3}. If x==i, y==j mod 4, where i,j are in R, then we write x<y mod 4, if 
i<j;  x>y mod 4, if  i>j. Let a(n) be the nth odious number, i.e., a(n)=A000069(n). We pose several conjectures about  S(n)=sum{i=1,...,n}a(i), n>=2.

Conjecture 1. If a(n-1)<a(n+1), a(n)<=a(n+2) mod 4, then S(n)=a(n)*a(n+1)/4.
Conjecture 2. If a(n-1)>a(n+1), a(n)>=a(n+2) mod 4, then S(n)=a(n)*a(n+1)/4+0.5.
Conjecture 3. If a(n-1)<a(n+1), a(n)>a(n+2) mod 4, then S(n)=a(n)*(a(n+1)-1)/4.
Conjecture  4. If a(n-1)>a(n+1), a(n)<a(n+2) mod 4, then S(n)=(a(n)+1)*a(n+1)/4.

Let b(n) be the nth evil number, i.e., b(n)=A001969(n), s(n)=sum{i=1,...,n}b(n). The following 4 conjectures are formulated by the same way with the replacement "a" by "b" and "S" by "s".

Are these conjectures indeed new?

Best regards,

 Shevelev Vladimir‎

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