[seqfan] "Odious-evil stability" of integers

[Vladimir Shevelev]

Dear Seqfans,
 
1) Let us understand under the index of "multiplicative odious-evil stability" of an odd positive integer t=2*n-1 the maximal number k(n) of consecutive odd numbers 2*i-1, 1<=i<=2*k(n) -1, such that all numbers t, 3*t, 5*t,...,(2*k(n)-1)*t are all odious, if t is odious, and are all evil, if t is evil.
Table of such indices begins  (n>=1) 1, 3, 2, 4, 4, 1, 1, 9, 8, 1, 1, 1, 1, 1, 1,16,...
One can prove that, if t is a Mersenne number  (A000225) , then index
 k >=(t+1)/2; besides, it is easy to prove that, if t-2 is a Mersenne numbers, then k>=(t-1)/2. Note also that, if t is odious number, such that
3*t is not in  A224072, then k=1.
2) It is interesting to research also "power stability" of odd integers>1.
For example, 3, 3^2, 3^3 are evil, while 3^4 is odious. So 3 has power stability of index 3. For 5 it is 1, for 7 it is 2, etc.
In connection with this one can propose the following interesting sequences (I give only formulations of them):  
1) a(n) is the minimal odd evil k, such that k^i, i=1,2,...,n, all are evil and a(n)=0, if there is no such k;
2) b(n) is the minimal odd odiousl k>1, such that k^i, i=1,2,...,n, all are odious and b(n)=0, if there is no such k.

 Best regards,
Vladimir

 Shevelev Vladimir‎