A238134 Number of primes p < n with q = floor((n-p)/4) and prime(q) - q + 1 both prime.

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

Offset

1,11

Comments

Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n such that q = floor((n-p)/m) and prime(q) - q + 1 are both prime.

In the cases m = 1, 2, this gives refinements of Goldbach's conjecture and Lemoine's conjecture (see also A235189). For m > 2, the conjecture is completely new.

See also A238701 for a similar conjecture involving primes q with q^2 - 2 also prime.

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(29) = 3 since 7, floor((29-7)/4) = 5 and prime(5) - 5 + 1 = 11 - 4 = 7 are all prime; 17, floor((29-17)/4) = 3 and prime(3) - 3 + 1 = 5 - 2 = 3 are all prime; 19, floor((29-19)/4) = 2 and prime(2) - 2 + 1 = 3 - 1 = 2 are all prime.

Mathematica

PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]
a[n_]:=Sum[If[PQ[Floor[(n-Prime[k])/4]], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A045917, A046927, A234694, A234695, A235189, A237705, A238701.

Sequence in context: A172397 A237815 A238701 * A143489 A349393 A341155

Adjacent sequences: A238131 A238132 A238133 * A238135 A238136 A238137

Keyword

nonn

Author

Zhi-Wei Sun, Mar 03 2014

Status

approved