A181188 - OEIS

%I #4 Mar 31 2012 10:24:04

%S 31,101,167,229,269,271,307,311,313,317,331,359,439,479,487,491,691,

%T 787,797,3739,3761,3821,4019,4093,4153,4231,4241,4243,4253,5839,5843,

%U 5857,5861,6367,6469,6473,6553,6637,6653,6673,6679,7121,7219,7297,7307,7309,7351,7561,7583,7603,7607,7681,8311

%N Primes at which the prime number race between the two prime classes with different sign of sin(prime(.)) changes leader.

%C Split the prime numbers into A070754 and A070753 according to the sign of the sine function:

%C  2,  3,  7, 13, 19| 47, 53, 59, 71, 83, 89, 97,101|103,107,109,127,139,151|179,191,197,223,...

%C  5, 11, 17, 23, 29| 31, 37, 41, 43, 61, 67, 73, 79|113,131,137,149,157,163|167,173,181,193,199,...

%C Comparison of A070754(i) with A070753(i) defines a prime number race. The leader chances at places i where sign( A070754(i)-A070753(i) ) <> sign( A070754(i+1)-A070753(i+1) ) indicated by the vertical bars above.

%C An equivalent observation is that the partial sum s(k) := sum_{i=1..k} A070748(i) has zeros at prime(k)= 29, 101, 163, 229, 263, 271,...

%C The sequence contains each prime(k+1) where s(k) >=0 and s(k+1)<0 or s(k) <0 and s(k+1)>=0. Cases where s(k) touches zero without actually flipping the sign are not relevant.

%H John Derbyshire and Mikhail Gaichenkov, <a href="http://www.johnderbyshire.com/Books/Prime/Blog/page.html#radians">The sign of the sine of p</a>

%p isA070753 := proc(n) is(sin(ithprime(n))<0) ; end proc:

%p A070748 := proc(n) option remember; if isA070753(n) then -1 ; else 1; end if; end proc:

%p A070748s := proc(n) add( A070748(i),i=1..n) ; end proc:

%p for n from 1 to 10000 do if A070748s(n) >= 0 and A070748s(n+1) < 0 or A070748s(n) <0 and A070748s(n+1) >= 0 then printf("%d,",ithprime(n+1)) ; end if;end do:

%o (PARI) s=0; p=0; while(1, p=nextprime(p+1); s+=(-1)^(p\Pi); if(s<=-7568,print1(p,", ")))

%o (PARI) s=0;forprime(p=2,2000,s+=(-1)^(p\Pi);print1(s,", "))

%K nonn

%O 1,1

%A _Mikhail Gaichenkov_, Oct 09 2010