OFFSET
1,6
COMMENTS
The first n such that a(n) = 5 is a(102). Records in a(n) are a(n) = A002496(n)+1. Hardy and Littlewood conjectured that, asymptotically, a(n) ~ c*(sqrt(n))/log n, where c ~ 1.3727.
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 2nd Edn., Springer, 1994, A1, pp.4-5.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
a(3) = 1 because the unique prime of the form k^2 + 1 less than 3 is 1^2 + 1 = 2. The smallest value of n to reach the next record is a(6) = 2 because a(18) = 2, the two primes of the form k^2 + 1 less than 6 are 2 and 2^2 + 1 = 5. The smallest value of n to reach the next record is a(18) = 3 because the three primes of the form k^2 + 1 less than 18 are 2, 5, and 4^2 + 1 = 17.
MATHEMATICA
Accumulate[Table[If[PrimeQ[n]&&IntegerQ[Sqrt[n-1]], 1, 0], {n, 0, 120}]] (* Harvey P. Dale, Jun 22 2024 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jun 28 2010
STATUS
approved