OFFSET
1,1
COMMENTS
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
KEYWORD
nonn
AUTHOR
Mikhail Gaichenkov, Oct 09 2010
STATUS
approved