cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A070751 Numbers n such that sin(n) < 0.

Original entry on oeis.org

4, 5, 6, 10, 11, 12, 16, 17, 18, 22, 23, 24, 25, 29, 30, 31, 35, 36, 37, 41, 42, 43, 48, 49, 50, 54, 55, 56, 60, 61, 62, 66, 67, 68, 69, 73, 74, 75, 79, 80, 81, 85, 86, 87, 92, 93, 94, 98, 99, 100, 104, 105, 106, 110, 111, 112, 113, 117
Offset: 1

Views

Author

Reinhard Zumkeller, May 04 2002

Keywords

Comments

A070747(a(n)) = -1.

Crossrefs

Cf. A070752, A070753.

Programs

  • Mathematica
    Select[Range[150],Sin[#]<0&] (* Harvey P. Dale, Dec 24 2023 *)

A070748 a(n) = signum(sin(prime(n))), where signum=A057427, prime=A000040.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, 1
Offset: 1

Views

Author

Reinhard Zumkeller, May 04 2002

Keywords

Links

Crossrefs

Cf. A070747, A070750, A070749, A070753, A070754.

Programs

A070754 Primes p such that sin(p) > 0.

Original entry on oeis.org

2, 3, 7, 13, 19, 47, 53, 59, 71, 83, 89, 97, 101, 103, 107, 109, 127, 139, 151, 179, 191, 197, 223, 227, 229, 233, 239, 241, 271, 277, 283, 311, 317, 347, 353, 359, 367, 373, 379, 397, 409, 421, 449, 461, 467, 479, 491, 499, 503
Offset: 1

Views

Author

Reinhard Zumkeller, May 04 2002

Keywords

Comments

A070748(A049084(a(n))) = A070747(a(n)) = 1.

Crossrefs

Cf. A070753, A070748, A002144 (sin((Pi/2)*p) > 0 instead of sin(p) > 0), A070752.

Programs

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

Original entry on oeis.org

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

Views

Author

Mikhail Gaichenkov, Oct 09 2010

Keywords

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

Programs

  • 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:
  • 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,", "))
Showing 1-4 of 4 results.

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