A115366 a(n) = the number of values of k <= 10^n such that sqrt(k*(k+1)*(k+2)*(k+3)+1) is prime.

1, 9, 50, 313, 2188, 17075, 139484, 1179766, 10220078, 90159466, 806928985, 7302511765

Offset

0,2

Comments

sqrt(k*(k+1)*(k+2)*(k+3)+1) = k^2 + 3*k + 1.

a(n)/A006880(n) ~= 1.78, 1.78, 1.7769, 1.7752, 1.7738, 1.7731. Conjecture: a(n)/A006880(n) -> 1.77...

Links

Table of n, a(n) for n=0..11.

Mathematica

c = 0; k = 1; Do[ While[k <= 10^n, If[ PrimeQ@ Round@ Sqrt[k(k + 1)(k + 2)(k + 3) + 1], c++ ]; k++ ]; Print@c, {n, 0, 9}] (* Robert G. Wilson v, May 01 2006 *)

Prog

(PARI) g(n)= { for(j=0, n, c=0; for(x=0, 10^j, y=round(sqrt(x*(x+1)*(x+2)*(x+3)+1)); if(ispseudoprime(y), c++)); print1(c", ") ) }
(Python)
from sympy import isprime
def A115366(n): return sum(1 for k in range(1, 10**n+1) if isprime(k*(k+3)+1)) # Chai Wah Wu, Jun 19 2024

Crossrefs

Cf. A006880.

Sequence in context: A231413 A007681 A279972 * A188210 A308646 A338510

Adjacent sequences: A115363 A115364 A115365 * A115367 A115368 A115369

Keyword

nonn,more

Author

Cino Hilliard, Mar 07 2006

Extensions

Edited by Don Reble, Apr 24 2006

a(10) from Michael S. Branicky, Jun 20 2024

a(11) from Michael S. Branicky, Jun 23 2024

Status

approved