A181188 Primes at which the prime number race between the two prime classes with different sign of sin(prime(.)) changes leader.

31, 101, 167, 229, 269, 271, 307, 311, 313, 317, 331, 359, 439, 479, 487, 491, 691, 787, 797, 3739, 3761, 3821, 4019, 4093, 4153, 4231, 4241, 4243, 4253, 5839, 5843, 5857, 5861, 6367, 6469, 6473, 6553, 6637, 6653, 6673, 6679, 7121, 7219, 7297, 7307, 7309, 7351, 7561, 7583, 7603, 7607, 7681, 8311

Offset

1,1

Comments

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

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

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

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.

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,...

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.

Links

Table of n, a(n) for n=1..53.

John Derbyshire and Mikhail Gaichenkov, The sign of the sine of p

Maple

isA070753 := proc(n) is(sin(ithprime(n))<0) ; end proc:
A070748 := proc(n) option remember; if isA070753(n) then -1 ; else 1; end if; end proc:
A070748s := proc(n) add( A070748(i), i=1..n) ; end proc:
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:

Prog

(PARI) s=0; p=0; while(1, p=nextprime(p+1); s+=(-1)^(p\Pi); if(s<=-7568, print1(p, ", ")))
(PARI) s=0; forprime(p=2, 2000, s+=(-1)^(p\Pi); print1(s, ", "))

Crossrefs

Sequence in context: A139509 A187517 A039463 * A323617 A142096 A268986

Adjacent sequences: A181185 A181186 A181187 * A181189 A181190 A181191

Keyword

nonn

Author

Mikhail Gaichenkov, Oct 09 2010

Status

approved