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A239428 Number of ordered ways to write n = k + m with 0 < k <= m such that pi(2*k) - pi(k) and pi(2*m) - pi(m) are both prime, where pi(x) denotes the number of primes not exceeding x. 3
0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 3, 2, 3, 3, 2, 4, 2, 4, 2, 2, 3, 1, 4, 2, 3, 3, 2, 4, 1, 5, 4, 5, 6, 4, 6, 4, 5, 4, 3, 5, 2, 4, 2, 2, 2, 1, 2, 1, 2, 3, 3, 2, 3, 3, 5, 6, 4, 6, 5, 7, 4, 5, 5, 4, 5, 3, 5, 6, 5, 6, 4, 6, 4, 6, 5, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

Conjecture: a(n) > 0 for all n > 9, and a(n) = 1 only for n = 8, 10, 11, 26, 33, 50, 52.

This implies that there are infinitely many positive integers n with pi(2*n) - pi(n) prime.

Recall that Bertrand's postulate proved by Chebyshev in 1850 asserts that pi(2*n) - pi(n) > 0 for all n > 0.

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(11) = 1 since 11 = 4 + 7 with pi(2*4) - pi(4) = 4 - 2 = 2 and pi(2*7) - pi(7) = 6 - 4 = 2 both prime.

a(26) = 1 since 26 = 13 + 13 with pi(2*13) - pi(13) = 9 - 6 = 3 prime.

a(33) = 1 since 33 = 6 + 27 with pi(2*6) - pi(6) = 5 - 3 = 2 and pi(2*27) - pi(27) = 16 - 9 = 7 both prime.

a(50) = 1 since 50 = 23 + 27 with pi(2*23) - pi(23) = 14 - 9 = 5 and pi(2*27) - pi(27) = 16 - 9 = 7 both prime.

a(52) = 1 since 52 = 21 + 31 with pi(2*21) - pi(21) = 13 - 8 = 5 and pi(2*31) - pi(31) = 18 - 11 = 7 both prime.

MATHEMATICA

p[n_]:=PrimeQ[PrimePi[2n]-PrimePi[n]]

a[n_]:=Sum[If[p[k]&&p[n-k], 1, 0], {k, 1, n/2}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000040, A000720, A108954.

Sequence in context: A106696 A131839 A143299 * A257497 A266225 A248596

Adjacent sequences:  A239425 A239426 A239427 * A239429 A239430 A239431

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 20 2014

STATUS

approved

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Last modified January 23 23:01 EST 2022. Contains 350515 sequences. (Running on oeis4.)