

A237496


Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m)  2 prime, where pi(.) is given by A000720.


6



0, 0, 0, 0, 0, 1, 2, 4, 4, 3, 2, 3, 3, 3, 5, 3, 1, 4, 5, 5, 7, 4, 1, 2, 1, 1, 1, 1, 1, 3, 6, 7, 8, 8, 8, 8, 8, 9, 11, 11, 11, 11, 9, 7, 7, 4, 1, 2, 1, 2, 3, 5, 7, 10, 14, 14, 14, 10, 6, 10, 14, 16, 19, 16, 13, 12, 11, 10, 7, 6, 5, 3, 3, 4, 3, 6, 9, 13, 17, 18
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OFFSET

1,7


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) Any integer n > 23 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) prime. Also, each integer n > 25 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m)  1 prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..3000


EXAMPLE

a(6) = 1 since 6 = 3 + 3 with pi(3) + pi(3)  2 = 2 + 2  2 = 2 prime.
a(17) = 1 since 17 = 2 + 15 with pi(2) + pi(15)  2 = 1 + 6  2 = 5 prime.
a(99) = 1 since 99 = 1 + 98 with pi(1) + pi(98)  2 = 0 + 25  2 = 23 prime.


MATHEMATICA

PQ[n_]:=n>0&&PrimeQ[n]
p[k_, m_]:=PQ[PrimePi[k]+PrimePi[m]2]
a[n_]:=Sum[If[p[k, nk], 1, 0], {k, 1, n/2}]
Table[a[n], {n, 1, 80}]


CROSSREFS

Cf. A000040, A000720, A232465, A237284, A237291, A237453, A237497.
Sequence in context: A185356 A008777 A306691 * A202690 A257978 A193358
Adjacent sequences: A237493 A237494 A237495 * A237497 A237498 A237499


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 08 2014


STATUS

approved



