[seqfan] Re: a(n)+ [the a(n)th digit of T] is odd (same idea)

M. F. Hasler seqfan at hasler.fr
Sat Feb 6 22:33:00 CET 2016


Hello Eric & SeqFans,

to avoid using decimal digits, I wondered whether we can replace "digit" by
"prime factor" in the sense of A027746 <https://oeis.org/A027746>,
in this and most others of your sequences ;
e.g., here again to have s(n) + f(n) always odd:

n = 1  2  3  4  *5*  6  *7*  8  9 10 11 12 13 14 15 16
F = 2, 1, 5, 7, 2, 2, 2, 5,11,13, 2, 2, 2,17, 2, 7
s = 2, 1, 5, 7, 4,10,11,13, 8,17,14, ...

s(1)=1 not possible.
*s(1)=2*=f(1).  [Final results are in boldface, I hope they reach you well.]
*s(2)=1*=f(2).
s(3)=3=f(3) not possible because 3+3=even.
s(3)=4=f(3) is not possible because this leads to f(4)=2
*s(3)=5*=f(3) is possible, we must require *f(5) = 2*.
We note that *3 can never occur* because 3+f(3) = 3+3 is not odd
s(4)=4 => f(4)=2 is not possible because 4+2 is not odd
s(4)=6 => f(5)=3 is not possible
*s(4)=7**=f(4)* is possible. we require that* f(7) = 2.*
s(5)=2q because *f(5)=2*,
*s(5)=4,* i.e., q=*2=f(6)* is ok since f(4)=7 is odd
We note that *6 can never occur* because 6+f(6) = 6+2 is not odd
s(6)=2q because *f(7)=2*
s(6)=6 not possible because f(6)=2 is not odd
s(6)=8 not possible because then f(8)=2 is not odd
*s(6)=10, f(8)=5* is possible. we must have* f(10)=odd*
s(7)=8 => f(9,10,11)=2  is not possible because f(10)=odd.
s(7)=9 => f(9,10)=3 is not possible (9+3 is even)
*s(7)=11= f(9)* is possible, we must have* f(11)=2*.
Now f(10) must be odd and f(11)=2 so s(8) must be an odd prime,
*s(8)**=13**=f(10)* is the smallest available. Thus* f(13)=2*.
s(9)=2q because f(11)=2 and we can use
*s(9)=8, f(11,12,13)=2*,  possible because 8+f(8)=13.
...
With the highly unoptimized PARI code below I get:
s = [2, 1, 5, 7, 4, 10, 11, 13, 8, 17, 14, 16, 15, 19, 22, 28, 27, 30, 34,
35, 36, 38, 39, 42, 43, 44, 46, 47, 48, 50, 51, 56, 57, 59, 64, 66, 67, 69,
70, 72, 75, 77, 79, 82, 84, 88, 89, 91, 92, 94, 96, 98, 103, 104, 107, 110,
112, 113, 119, 121, 126, 129, 132, ...]
Numbers which will never occur in this sequence:
[3, 6, 9, 12, 18, 20, 21, 23, 24, 25, 26, 29, 31, 32, 33, 37, 40, 41, 45,
...]

Happy carnival and beware of zika virus ;-) !

Maximilian

(PARI)
check(s,f=concat(apply(A027746 <https://oeis.org/A027746>_row,s)
))=!for(i=1,#s,s[i]<=#f&&(bittest(s[i]+f[s[i]],0)||return))
s=[2];for(n=1,99,S=Set(s);s=concat(s,1);for(k=1,S[#S]+99,!setsearch(S,k)&&(s[#s]=k)&&check(s)&&break))


On Sat, Feb 6, 2016 at 1:45 PM, Eric Angelini <Eric.Angelini at kntv.be> wrote:

>
> "a(n)+ [the a(n)th digit of T] is odd"
> (To extend T always use the smallest
> available integer not yet in T and not
> leading to a contradiction.)
>



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