A127977 The minimum excess in the prime race of odious primes versus evil primes in the interval (2^(n-1),2^n).

0, 1, 4, 7, 13, 19, 39, 53, 104, 138, 251, 334, 590, 715, 1353, 1855, 3659, 5221, 10484, 14933, 27491, 35474, 68816, 97342, 186405, 265255

Offset: 5

Author

Jonathan Vos Post, Jun 07 2007

Keywords

• nonn
• base

Comments

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.

Examples

			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,...
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.
		

Links

• Vladimir Shevelev, A Conjecture on Primes and a Step towards Justification, arXiv:0706.0786 [math.NT], 2007. See table 1, p. 2.
• Vladimir Shevelev, On excess of the odious primes, arXiv:0707.1761 [math.NT], 2007.

Crossrefs

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

Programs

Maple

read("transforms") ; # see oeis.org/transforms.txt
isA000069 := proc(n) type(wt(n),'odd') ; end proc;
isA027697 := proc(n) isprime(n) and isA000069(n) ; end proc:
isA027699 := proc(n) isprime(n) and not isA000069(n) ; end proc:
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:
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:
oddPi := proc(n) odiPi(n)-eviPi(n) ; end proc:
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:
for n from 5 do print(n,A127977(n)) ; end do; # R. J. Mathar, Sep 03 2011

Mathematica

wt[n_] := DigitCount[n, 2, 1];
isA000069[n_] := OddQ[wt[n]];
isA027697[n_] := PrimeQ[n] && isA000069[n];
isA027699[n_] := PrimeQ[n] && !isA000069[n];
odiPi[n_] := odiPi[n] = If[n==0, 0, an1 = odiPi[n-1]; If[isA027697[n], an1+1, an1]];
eviPi[n_] := eviPi[n] = If[n==0, 0, an1 = eviPi[n-1]; If[isA027699[n], an1+1, an1]];
oddPi[n_] := odiPi[n] - eviPi[n];
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];
Table[an = A127977[n]; Print[an]; an, {n, 5, 30}] (* Jean-François Alcover, Jan 23 2018, after R. J. Mathar *)

Extensions

Published numbers corrected and checked up to n=23 by R. J. Mathar, Sep 03 2011