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A239430 Number of ways to write n = k + m with k > 0 and m > 0 such that pi(2*k) - pi(k) is prime and pi(2*m) - pi(m) is a square, where pi(x) denotes the number of primes not exceeding x. 2
0, 0, 0, 0, 1, 1, 2, 2, 4, 3, 3, 3, 2, 4, 2, 5, 3, 4, 5, 1, 5, 3, 6, 7, 5, 9, 3, 7, 5, 4, 7, 5, 9, 5, 5, 4, 2, 4, 2, 5, 4, 6, 7, 5, 9, 6, 9, 8, 7, 10, 8, 10, 6, 7, 6, 6, 7, 6, 5, 6, 7, 5, 5, 6, 7, 8, 7, 10, 11, 12, 11, 7, 6, 9, 10, 8, 7, 6, 7, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 20.

LINKS

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

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

EXAMPLE

a(5) = 1 since 5 = 4 + 1 with pi(2*4) - pi(4) = 4 - 2 = 2 prime and pi(2*1) - pi(1) = 1^2.

a(20) = 1 since 20 = 8 + 12 with pi(2*8) - pi(8) = 6 - 4 = 2 prime and pi(2*12) - pi(12) = 9 - 5 = 2^2.

MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]

s[n_]:=SQ[PrimePi[2n]-PrimePi[n]]

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

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

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

CROSSREFS

Cf. A000040, A000290, A000720, A239428.

Sequence in context: A122687 A002948 A117113 * A260951 A240600 A227183

Adjacent sequences:  A239427 A239428 A239429 * A239431 A239432 A239433

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 20 2014

STATUS

approved

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Last modified September 15 14:38 EDT 2019. Contains 327078 sequences. (Running on oeis4.)