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A181188
Primes at which the prime number race between the two prime classes with different sign of sin(prime(.)) changes leader.
0
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
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
KEYWORD
nonn
AUTHOR
Mikhail Gaichenkov, Oct 09 2010
STATUS
approved