A238585 Number of primes p < n with prime(p)^2 + (prime(n)-1)^2 prime.

0, 0, 0, 1, 1, 0, 1, 2, 2, 1, 1, 1, 3, 2, 3, 2, 2, 3, 1, 5, 1, 1, 3, 2, 4, 5, 2, 4, 3, 4, 1, 4, 5, 3, 4, 6, 3, 2, 2, 2, 2, 1, 8, 1, 3, 4, 7, 2, 5, 3, 2, 2, 4, 7, 4, 3, 2, 3, 5, 7, 5, 3, 6, 6, 5, 3, 4, 5, 2, 2, 2, 3, 7, 2, 3, 7, 3, 4, 10, 3

Offset

1,8

Comments

Conjecture: (i) a(n) > 0 unless n divides 6, and a(n) = 1 only for n = 4, 5, 7, 10, 11, 12, 19, 21, 22, 31, 42, 44.

(ii) If n > 2 is not equal to 9, then prime(n)^2 + (prime(p) - 1)^2 is prime for some prime p < n.

(iii) For n > 3, there is a prime p < n with prime(p) + prime(n) + 1 prime. If n > 9 is not equal to 18, then prime(p)^2 + prime(n)^2 - 1 is prime for some prime p < n.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Example

a(7) = 1 since 3 and prime(3)^2 + (prime(7)-1)^2 = 5^2 + 16^2 = 281 are both prime.
a(44) = 1 since 23 and prime(23)^2 + (prime(44)-1)^2 = 83^2 + 192^2 = 43753 are both prime.

Mathematica

p[n_, k_]:=PrimeQ[k]&&PrimeQ[Prime[k]^2+(Prime[n]-1)^2]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A232465, A238580.

Sequence in context: A182591 A321393 A058914 * A123682 A134513 A275648

Adjacent sequences: A238582 A238583 A238584 * A238586 A238587 A238588

Keyword

nonn

Author

Zhi-Wei Sun, Mar 01 2014

Status

approved