[seqfan] A hypothetical sequence

Vladimir Shevelev shevelev at bgu.ac.il
Wed Sep 7 15:31:43 CEST 2011

In 1969, D.J.Newman proved that in the race of odious multiples of 3  versus evil multiples of 3 the latter permanently win from the very outset  (sequences of odious and evil numbers are A000069, A001969).
Denote by V_p the sequence of positive integers with the smallest prime divisor p. Note that the Newman result is true for V_3. For p>3, I proved (2008) that, for sufficiently large n (n>n_p), the race of odious numbers from  V_p versus evil multiples from V_p  the latter lose. A very interesting question: does there exist a prime p>3, for which they permanently not win from the very outset?
  Unfortunately, n_p can be very large and I have no an upper estimate for it. Nevertheless, I conjecture that the question is answered in affirmative. Moreover, I conjecture that the first such a prime is 11. With help of some numerical observations I did  conjectures that in this series there are also 19, 41, 67, 107, 173, 179, 181, 307, 313, 421, 431, 433, 587, 601, 631, 641, 647, 727, 787. Up to 1000, I did not find other such primes. I wonder if  a sequence with only  hypothetical terms can be submitted to OEIS? Can anyone verify and, maybe, remove any term?

Best regards,

 Shevelev Vladimir‎

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