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A232443 Number of ways to write n = p + q - pi(q), where p and q are odd primes, and pi(q) is the number of primes not exceeding q. 8
0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 4, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 2, 2, 5, 4, 5, 5, 3, 3, 4, 4, 4, 5, 4, 2, 6, 5, 5, 5, 3, 4, 8, 5, 5, 6, 2, 4, 7, 6, 6, 5, 3, 5, 7, 6, 7, 6, 4, 6, 6, 5, 7, 5, 6, 6, 7, 7, 7, 6, 4, 5, 7, 8, 8, 7, 7, 7, 7, 8, 9, 5, 6, 9, 9, 7, 6, 7, 6, 7, 8, 3, 8, 9, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 3. Moreover, every n = 6, 7, ... can be written as p + q - pi(q) with p, p + 6 and q all prime.

(ii) For each integer n > 7, there is a prime p < n with n + p - pi(p) prime.

(iii) Any integer n > 4 not equal to 9 or 17 can be written as p + q + pi(q) with p and q both prime.

(iv) Each integer n > 7 can be written as p + q + pi(p) + pi(q) with p and q both prime.

LINKS

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

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

EXAMPLE

a(4) = 1 since 4 = 3 + 3 - pi(3) with 3 prime.

a(5) = 1 since 5 = 3 + 5 - pi(5) with 3 and 5 prime.

a(6) = 2 since 6 = 3 + 7 - pi(7) = 5 + 3 - pi(3) with 3, 5, 7 all prime.

a(7) = 1 since 7 = 5 + 5 - pi(5) with 5 prime.

a(11) = 1 since 11 = 5 + 11 - pi(11) with 5 and 11 both prime.

MATHEMATICA

PQ[n_]:=PQ[n]=n>2&&PrimeQ[n]

a[n_]:=Sum[If[PQ[n-Prime[k]+k], 1, 0], {k, 2, PrimePi[2n-2]}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000040, A000720, A002375, A232398.

Sequence in context: A213325 A043555 A242753 * A118821 A118824 A209402

Adjacent sequences:  A232440 A232441 A232442 * A232444 A232445 A232446

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 23 2013

STATUS

approved

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Last modified June 25 03:50 EDT 2019. Contains 324338 sequences. (Running on oeis4.)