

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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei 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[nk], 1, 0], {k, 1, n1}]
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

ZhiWei Sun, Mar 20 2014


STATUS

approved



