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.
For the fourth term q is 5. For primes 2,3,5 and 7 the mod 5 values are 2,3,0 and 2 respectively, so there is no change in the race. For the next prime 11, mod 5 gives 1, q*k+1 now leads 1 to 0, and the race is over.
/* C language program used to investigate prime number races. Computes the first lead of qn+1 over qn-1 for q from 2 to 999. By Andy Martin oldadit@gmail.com 8/12/2016. Requires Kim Walisch's primesieve library from http://primesieve.org Iteration based on the primesieve_iterator.c example. */ #include#include #include #define UPDATE_COUNT 10000000000ull void race(uint64_t q) { uint64_t prime = 0; uint64_t m1 = 0; uint64_t m_1 = 0; uint64_t rem = 0; uint64_t update = UPDATE_COUNT; primesieve_iterator pi; primesieve_init(&pi); while (prime = primesieve_next_prime(&pi)) { if ((rem = prime % q) == 1){ m1 += 1; } else if (rem == q-1) { m_1 += 1; } if (m1 > m_1){ printf("Race mod %3llu ends at %12llu with %11llu pi(x;%llu,1) and %11llu pi(x;%llu,%llu)\n", q, prime, m1, q, m_1, q, q-1); break; } /* Enable for update on long races where q = 3,6,8,12,24,168 */ if (prime > update) { printf(" Race mod %llu ongoing at prime %llu with m1 %llu and m_1 %llu diff: %llu\n", q, prime, m1, m_1, m_1 - m1); update += UPDATE_COUNT; } } primesieve_free_iterator(&pi); } int main() { uint64_t i; for(i=2; i<1000; i++){ race(i); } return(0); }
from sympy import nextprime; p, r1, r5 = 1, 0, 0
for n in range(1, 30734263):
p = nextprime(p); r = p%8
if r == 1: r1 += 1
elif r == 5: r5 += 1
if r in {1, 5} and r1 == r5 + 1: print(n, end = ', ') # Ya-Ping Lu, Jan 08 2025
c1:= 0; c3:= 0: p:= 2: count:= 0: Res:= NULL:
while count < 100 do
p:= nextprime(p);
if c1 = c3 - 2 then
count:= count+1;
Res:= Res, p;
fi;
if p mod 4 = 1 then c1:=c1+1
else c3:= c3+1
fi
od:
Res; # Robert Israel, Nov 07 2018
a(5) = 8 as a(4) = 6 and 8 is unused and shares a factor with 6. Note that 3 cannot be chosen as 3 is of the form 4*k + 3, and no primes of form 4*k + 1 have yet occurred. This is the first term to differ from A064413. a(7) = 5 as a(6) = 10 and 5 is unused and shares a factor with 10. This is the first prime of the form 4*k + 1 to occur. a(9) = 3 as a(8) = 15 and 3 is unused and shares a factor with 15. As a prime of the form 4*k + 1 has occurred, one of the form 4*k + 3 is now allowed.
a(1)=26829 is the first odd composite 1 mod 4 where the count 5238 is the same for 26835 3 mod 4
a(1)=26835 is the first odd composite 3 mod 4 where the count 5238 is the same for 26829 1 mod 4
At 26829 1 mod 4 and 26835 3 mod 4, the count of odd composites is equal for each run at 5238; so a(1)=5238. [Compare to the prime 26833 1 mod 4 where equality occurs at count 1471 and the first reversal in the race occurs at 26861.]
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