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A308649
Primes prime(k) such that (prime(k), prime(k+1)), (prime(k+2), prime(k+3)), (prime(k+4), prime(k+5)) form a triangle of area 2.
1
59, 739, 1601, 2777, 3041, 3307, 3911, 3917, 4787, 4987, 6199, 6317, 6959, 7669, 8923, 10151, 12491, 14009, 14423, 14737, 15787, 16229, 16691, 17657, 24509, 27737, 28813, 30169, 32771, 34667, 35267, 36767, 39749, 40433, 41641, 44089, 44267, 45959, 52057, 57059, 58913, 60611, 60919, 61631, 62723
OFFSET
1,1
COMMENTS
If prime(k) is in the sequence, then so is prime(j) if prime(j+2)-prime(j) = prime(k+2)-prime(k), prime(j+4)-prime(j) = prime(k+4)-prime(k), prime(j+3)-prime(j+1)=prime(k+3)-prime(k+1) and prime(j+5)-prime(j+1) = prime(k+5)-prime(k+1).
Dickson's conjecture implies that the sequence is infinite.
LINKS
EXAMPLE
a(3) = 1601 is in the sequence because 1601, 1607, 1609, 1613, 1619, 1621 are consecutive primes and the triangle with vertices (1601, 1607), (1609, 1613), (1619, 1621) has area 2.
MAPLE
P:= map(ithprime, [$2..10000]):
g:= (t1, t2, t3, t4, t5) -> t4*(t3-t1)-t2*(t5-t1):
P[select(i -> abs(g(seq(P[i+j]-P[i], j=1..5)))=4, [$1..9994])];
PROG
(PARI) area(u, v, w) = abs((u[1]-w[1])*(v[2]-u[2])-(u[1]-v[1])*(w[2]-u[2]))/2
is(p) = my(i=primepi(p), v=primes([p, prime(i+5)])); area([v[1], v[2]], [v[3], v[4]], [v[5], v[6]])==2
forprime(p=1, 63000, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Jun 13 2019
CROSSREFS
Sequence in context: A140024 A219064 A263127 * A274503 A142363 A215433
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Jun 13 2019
STATUS
approved