

A237720


Number of primes p <= (n+1)/2 with floor( sqrt(np) ) prime.


4



0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 3, 4, 3, 4, 4, 4, 3, 4, 3, 3, 4, 5, 4, 5, 4, 5, 6, 6, 5, 6, 7, 8, 8, 8, 7, 7, 5, 6, 5, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,7


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 5, and a(n) = 1 only for n = 6, 23, 24, 111, 112, ..., 120.
(ii) For any integer n > 2, there is a prime p < n with floor(sqrt(n+p)) prime.
Note that floor(sqrt(n)) is the number of squares among 1, ..., n.
See also A237705, A237706 and A237721 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(6) = 1 since 2 and floor(sqrt(62)) = 2 are both prime.
a(23) = 1 since 11 and floor(sqrt(2311)) = 3 are both prime.
a(24) = 1 since 11 and floor(sqrt(2411)) = 3 are both prime.
a(27) = 2 since 2 and floor(sqrt(272)) = 5 are both prime, and 13 and floor(sqrt(2713)) = 3 are both prime.
a(n) = 1 for n = 111, ..., 116 since 53 and floor(sqrt(n53)) = 7 are both prime.
a(n) = 1 for n = 117, 118, 119, 120 since 59 and floor(sqrt(n59)) = 7 are both prime.


MATHEMATICA

q[n_]:=PrimeQ[Floor[Sqrt[n]]]
a[n_]:=Sum[If[q[nPrime[k]], 1, 0], {k, 1, PrimePi[(n+1)/2]}]
Table[a[n], {n, 1, 70}]


CROSSREFS

Cf. A000040, A000290, A237706, A237710, A237721.
Sequence in context: A071673 A174199 A072660 * A075172 A347643 A237261
Adjacent sequences: A237717 A237718 A237719 * A237721 A237722 A237723


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 12 2014


STATUS

approved



