A070748 -id:A070748 - cp's OEIS frontend

A333688 Partial sums of A070748.

1, 2, 1, 2, 1, 2, 1, 2, 1, 0, -1, -2, -3, -4, -3, -2, -1, -2, -3, -2, -3, -4, -3, -2, -1, 0, 1, 2, 3, 2, 3, 2, 1, 2, 1, 2, 1, 0, -1, -2, -1, -2, -1, -2, -1, -2, -3, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, -2, -1, -2

Offset: 1

Bence Bernáth, Apr 02 2020

sign

This sequence counts what the prime number distribution is in the intervals where the sine function gives different signs: if a(n) is positive, it means that up to n more primes fall into the interval (2k*Pi, (2k+1)*Pi) than in ((2k+1)*Pi, (2k+2)*Pi) for k=0,1,2,3... When a(n) is zero, the first n primes are distributed equally between these intervals.

			For n=4, a(4) = signum(sin(2)) + signum(sin(3)) + signum(sin(5)) + signum(sin(7)) = 1 + 1 - 1 + 1 = 2.

Bence Bernáth, Table of n, a(n) for n = 1..10000

Cf. A007504, A070748.

primes_up_to=1000;
sequence(1)=1;
for n=2:1:primes_up_to
        if isprime(n)
            sequence(numel(primes(n)))=sum(sign(sin(primes(n))));
        end
end
result=transpose((sequence));

Accumulate @ Table[Sign @ Sin @ Prime[i], {i, 1, 70}] (* Amiram Eldar, Apr 02 2020 *)

a(n) = sum(k=1, n, sign(sin(prime(k)))); \\ Michel Marcus, May 03 2020

a(n) = Sum_{k=1..n} A070748(k). - Sean A. Irvine, May 02 2020

A070750 0 if n-th prime is even, 1 if n-th prime is == 1 (mod 4), and -1 if n-th prime is == 3 (mod 4).

Original entry on oeis.org

0, -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

Reinhard Zumkeller, May 04 2002

Also, sin(prime(n)*Pi/2), where prime(n) = A000040(n), Pi=3.1415... (original definition).
Also imaginary part of primes mapped as defined in A076340, A076341: a(n) = A076341(A000040(n)), real part = A076342.
Legendre symbol (-1/prime(n)) for n > 1. - T. D. Noe, Nov 05 2003
For n > 1, let p = prime(n) and m = (p-1)/2. Then c(m) - a(n) == 0 (mod p), where c(m) = (2*m)!/(m!)^2 = A000984(m) is the central binomial coefficient. [Proof: By definition, c(m)*(m!)^2 - (p-1)! = 0 and therefore c(m)*(m!)^2*(-1)^(m+1) - (p-1)!*(-1)^(m+1) = 0. Now apply Wilson's theorem, (p-1)! == 1 (mod p), and its corollary, (m!)^2 == (-1)^(m+1) (mod p), and finally use the formula by T. D. Noe listed below to replace (-1)^m with a(n).] Similarly, C_m - 2*a(n) == 0 (mod p), with C_m = A000108(m) being the m-th Catalan number. [Proof: By definition, C_m*(p+1)*(m!)^2 - 2*(p-1)! = 0. The result follows proceeding as in the first proof.] - Stanislav Sykora, Aug 11 2014

			p = 4*k+1 (see A002144): a(p) = sin((4*k+1)*Pi/2) = sin(2*k*Pi + Pi/2) = sin(Pi/2) = 1.
p = 4*k+3 (see A002145): a(p) = sin((4*k+3)*Pi/2) = sin(2*k*Pi + 3*Pi/2) = sin(3*Pi/2) = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre Symbol.
Wikipedia, Wilson's theorem.

Cf. A000040, A039702, A070748, A070749, A002144, A002145, A000108, A000984, A134323, A257834, A076340, A076341.

Cf. A060294, A088538.

a070750 = (2 -) . (`mod` 4) . a000040  -- Reinhard Zumkeller, Feb 28 2012

a[n_] := JacobiSymbol[-1, Prime[n]]; a[1] = 0; Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Oct 05 2012, after T. D. Noe *)
Table[Which[EvenQ[p],0,Mod[p,4]==1,1,True,-1],{p,Prime[Range[80]]}] (* Harvey P. Dale, Mar 16 2020 *)

apply(n->2-n%4,primes(100)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = 2 - prime(n) mod 4 = 2 - A039702(n).
a(n) = (-1)^((prime(n)-1)/2) for n > 1. - T. D. Noe, Nov 05 2003
From Amiram Eldar, Dec 24 2022: (Start)
Product_{n>=1} (1 - a(n)/prime(n)) = 4/Pi (A088538).
Product_{n>=1} (1 + a(n)/prime(n)) = 2/Pi (A060294). (End)

Wording of definition changed by N. J. A. Sloane, Jun 21 2015

A070747 a(n) = signum(sin(n)), where signum=A057427.

Original entry on oeis.org

0, 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: 0

Reinhard Zumkeller, May 04 2002

a(n) <> 0 for n>0.

			For n=10: sin(10) = sin(10-2*Pi) < 0, as Pi < 10-2*Pi < 2*Pi, therefore a(10) = signum(sin(10)) = -1.

Cf. A000493, A000494, A032615, A070748, A070751, A070752.

a(n)=(-1)^(n/Pi%2\1) \\ Charles R Greathouse IV, Jun 13 2013

a(n) = if (n==0, 0, (-1)^(n\Pi%2)); \\ Michel Marcus, Mar 20 2021

a(n) = (-1)^A032615(n) for n>0. - Michel Marcus, Mar 20 2021

A070753 Primes p such that sin(p) < 0.

Original entry on oeis.org

5, 11, 17, 23, 29, 31, 37, 41, 43, 61, 67, 73, 79, 113, 131, 137, 149, 157, 163, 167, 173, 181, 193, 199, 211, 251, 257, 263, 269, 281, 293, 307, 313, 331, 337, 349, 383, 389, 401, 419, 431, 433, 439, 443, 457, 463, 487, 521

Offset: 1

Reinhard Zumkeller, May 04 2002

nonn

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

Cf. A070754, A070748, A002145 (sin((Pi/2)*p) < 0 instead of sin(p) < 0), A070751.

Select[Prime[Range[200]],Sin[#]<0&] (* Harvey P. Dale, May 07 2012 *)

select(p->p/Pi%2\1,primes(100)) \\ Charles R Greathouse IV, Jun 13 2013

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

Reinhard Zumkeller, May 04 2002

nonn

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

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

select(p->1-p/Pi%2\1,primes(100)) \\ Charles R Greathouse IV, Jun 13 2013

A070749 Nearest integer to sin(prime(n)), prime=A000040.

Original entry on oeis.org

1, 0, -1, 1, -1, 0, -1, 0, -1, -1, 0, -1, 0, -1, 0, 0, 1, -1, -1, 1, -1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, -1, -1, 1, -1, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, -1, 0, 0, 1, 0, 0, 0, 1, 0, -1, -1, -1, 1, 1, -1, 0, -1, -1, 0, -1, 0, -1, -1, 1, 0, 1, 1, 1, 1, 1, 0, -1, 1, -1, 1, -1, 0, -1, -1, -1, 0, 0, -1, 1, -1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, -1, -1, 0, -1, -1, 0, 1, 1

Offset: 1

Reinhard Zumkeller, May 04 2002

Cf. A070748, A000494, A070750.

More terms from Matthew Conroy, Jan 16 2006

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

Mikhail Gaichenkov, Oct 09 2010

nonn

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.

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

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:

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

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

cp's OEIS Frontend

A333688 Partial sums of A070748.

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A070750 0 if n-th prime is even, 1 if n-th prime is == 1 (mod 4), and -1 if n-th prime is == 3 (mod 4).

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A070747 a(n) = signum(sin(n)), where signum=A057427.

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A070753 Primes p such that sin(p) < 0.

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

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A070749 Nearest integer to sin(prime(n)), prime=A000040.

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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.

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