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

1,2

Comments

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.

Links

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

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A007504, A070748.

Sequence in context: A341945 A280535 A221171 * A319610 A288739 A111621

Adjacent sequences: A333685 A333686 A333687 * A333689 A333690 A333691

Keyword

sign

Author

Bence Bernáth, Apr 02 2020

Status

approved