

A233529


a(n) = {0 < k <= n/2: prime(k)*prime(nk)  6 is prime}.


2



0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 1, 4, 5, 1, 5, 3, 2, 1, 2, 5, 5, 4, 5, 6, 5, 5, 4, 8, 5, 7, 4, 3, 6, 6, 4, 8, 6, 7, 7, 8, 7, 5, 5, 5, 7, 8, 6, 13, 9, 5, 3, 9, 6, 8, 11, 5, 9, 9, 10, 8, 9, 14, 9, 10, 13, 11, 6, 9, 12, 10, 12, 14, 10, 12, 7, 13, 9, 7, 7, 15, 12, 6, 10, 11, 12, 12, 9, 18, 15, 14, 11, 10, 10, 8, 13, 21, 9, 14
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,9


COMMENTS

Conjectures:
(i) a(n) > 0 for all n > 5. Also, for any n > 5, 2*prime(k)*prime(nk)  3 is prime for some 0 < k < n.
(ii) For any n > 1 not among 3, 9, 13, 26, there is a positive integer k < n with prime(k)*prime(nk)  2 prime. For any n > 2 not among 8, 23, 33, there is a positive integer k < n with prime(k)*prime(nk)  4 prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014


EXAMPLE

a(8) = 1 since prime(4)*prime(4)  6 = 7*7  6 = 43 is prime.
a(10) = 1 since prime(3)*prime(7)  6 = 5*17  6 = 79 is prime.
a(16) = 1 since prime(3)*prime(13)  6 = 5*41  6 = 199 is prime.
a(20) = 1 since prime(7)*prime(13)  6 = 17*41  6 = 691 is prime.


MATHEMATICA

PQ[n_]:=n>0&&PrimeQ[n]
a[n_]:=Sum[If[PQ[Prime[k]*Prime[nk]6], 1, 0], {k, 1, n/2}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A232465, A232502, A232861, A233150, A233204, A233206, A233439.
Sequence in context: A187002 A177226 A059026 * A104471 A174828 A305309
Adjacent sequences: A233526 A233527 A233528 * A233530 A233531 A233532


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 11 2013


STATUS

approved



