A309655 The smallest possible nonnegative difference between the sum of the first n primes (A007504) and the sum of any number of the directly following and consecutive primes.

0, 2, 0, 3, 6, 15, 5, 16, 25, 3, 20, 39, 13, 36, 61, 17, 50, 6, 39, 76, 14, 53, 102, 28, 75, 132, 46, 101, 158, 46, 99, 174, 64, 145, 27, 114, 193, 51, 144, 239, 93, 194, 24, 135, 244, 74, 179, 294, 116, 253, 43, 162, 291, 61, 196, 337, 101, 250, 395, 139, 282, 427, 149, 324

Offset

0,2

Comments

Conjecture: a(0)=0, a(2)=0 and a(532)=0 are the only zeros in the sequence. a(n) has been computed for primes < 10^10. - Bob Andriesse, Oct 07 2020

Links

David Radcliffe, Table of n, a(n) for n = 0..10000

Example

a(2) = 2 + 3 - 5 = 0;
a(3) = 2 + 3 + 5 - 7 = 3;
a(6) = 2 + 3 + 5 + 7 + 11 + 13 - (17 + 19) = 5.
a(532)=0 because A007504(733) = 2*A007504(532).

Prog

(Python)
#Lists a(1)...a(100)
from sympy import prime
sumP=0
for i in range(1, 101):
  sumP+=prime(i)
  j=i+1
  diff=sumP
  while diff-prime(j) >=0:
   diff-=prime(j)
   j+=1
  print(diff, end=', ')
(Python)
from itertools import islice
from gmpy2 import next_prime
def a309655_gen():
    lower_prime = upper_prime = difference = 0
    while True:
        while difference >= 0:
            upper_prime = next_prime(upper_prime)
            if difference < upper_prime:
                yield int(difference)
            difference -= upper_prime
        lower_prime = next_prime(lower_prime)
        difference += 2 * lower_prime
print(list(islice(a309655_gen(), 64))) # David Radcliffe, Jun 10 2025

Crossrefs

Cf. A007504, A309714.

Sequence in context: A248966 A021495 A332356 * A260591 A127160 A334847

Adjacent sequences: A309652 A309653 A309654 * A309656 A309657 A309658

Keyword

nonn,look

Author

Bob Andriesse, Aug 11 2019

Status

approved