A127977 - OEIS

%I #24 Apr 18 2018 04:55:02

%S 0,1,4,7,13,19,39,53,104,138,251,334,590,715,1353,1855,3659,5221,

%T 10484,14933,27491,35474,68816,97342,186405,265255

%N The minimum excess in the prime race of odious primes versus evil primes in the interval (2^(n-1),2^n).

%C Shevelev conjectures (p.2) that for all natural numbers n other than 5 and 6, the number of evil primes not exceeding n <= the number of odious primes not exceeding n. Odious primes are A027697. Evil primes are A027699.

%H Vladimir Shevelev, <a href="https://arxiv.org/abs/0706.0786">A Conjecture on Primes and a Step towards Justification</a>, arXiv:0706.0786 [math.NT], 2007. See table 1, p. 2.

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0707.1761">On excess of the odious primes</a>, arXiv:0707.1761 [math.NT], 2007.

%e OdiPrimePi(x) for x >= 32 is 6, 6, 6, 6, 6, 7, 7, 7, 7, 8,.. and EviPrimePi(x) for x>=32 is 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6,...

%e The difference OdiPrimePi(x)-EviPrimePi(x) is 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3,.. so the minimum of the difference in the interval 2^(6-1)..2^6 is 1, yielding a(6)=1.

%p read("transforms") ; # see oeis.org/transforms.txt

%p isA000069 := proc(n) type(wt(n),'odd') ; end proc;

%p isA027697 := proc(n) isprime(n) and isA000069(n) ; end proc:

%p isA027699 := proc(n) isprime(n) and not isA000069(n) ; end proc:

%p odiPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027697(n) then an1+1 ; else an1 ; end if; end if; end proc:

%p eviPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027699(n) then an1+1 ; else an1 ; end if; end if; end proc:

%p oddPi := proc(n) odiPi(n)-eviPi(n) ; end proc:

%p A127977 := proc(n) local a,x ; a := 2^(n+1) ; for x from 2^(n-1)+1 to 2^n-1 do a := min(a,oddPi(x)) ; end do: a; end proc:

%p for n from 5 do print(n,A127977(n)) ; end do; # _R. J. Mathar_, Sep 03 2011

%t wt[n_] := DigitCount[n, 2, 1];

%t isA000069[n_] := OddQ[wt[n]];

%t isA027697[n_] := PrimeQ[n] && isA000069[n];

%t isA027699[n_] := PrimeQ[n] && !isA000069[n];

%t odiPi[n_] := odiPi[n] = If[n==0, 0, an1 = odiPi[n-1]; If[isA027697[n], an1+1, an1]];

%t eviPi[n_] := eviPi[n] = If[n==0, 0, an1 = eviPi[n-1]; If[isA027699[n], an1+1, an1]];

%t oddPi[n_] := odiPi[n] - eviPi[n];

%t A127977[n_] := Module[{a, x}, a = 2^(n+1); For[x = 2^(n-1)+1, x <= 2^n-1, x++, a = Min[a, oddPi[x]]]; a];

%t Table[an = A127977[n]; Print[an]; an, {n, 5, 30}] (* _Jean-François Alcover_, Jan 23 2018, after _R. J. Mathar_ *)

%Y Cf. A000069, A001969, A027697, A027699, A095005, A095006.

%K nonn,base

%O 5,3

%A _Jonathan Vos Post_, Jun 07 2007

%E Published numbers corrected and checked up to n=23 by _R. J. Mathar_, Sep 03 2011