

A237721


Number of primes p <= n with floor( sqrt(np) ) a square.


4



0, 1, 2, 2, 3, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 4, 4, 4, 4, 5, 5, 5, 5, 4, 3, 5, 4, 5, 4, 4, 4, 4, 3, 4, 3, 4, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3, 3, 2, 2, 4, 3, 3, 2, 2, 2, 4, 4, 5, 4, 4, 4, 3, 2, 3, 2, 3, 3
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 9, 10, 11, 12, 15, 16, 17.
We have verified this for n up to 10^6.
See also A237705, A237706 and A237720 for similar conjectures.


LINKS

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


EXAMPLE

a(2) = 1 since 2 is prime with floor(sqrt(22)) = 0^2.
a(3) = 2 since 2 is prime with floor(sqrt(32)) = 1^2, and 3 is prime with floor(sqrt(33)) = 0^2.
a(9) = a(10) = 1 since 7 is prime with floor(sqrt(97)) = floor(sqrt(107)) = 1^2.
a(11) = 1 since 11 is prime with floor(sqrt(1111)) = 0^2.
a(12) = 1 since 11 is prime with floor(sqrt(1211)) = 1^2.
a(15) = a(16) = 1 since 13 is prime with floor(sqrt(1513)) = floor(sqrt(1613)) = 1^2.
a(17) = 1 since 17 is prime with floor(sqrt(1717)) = 0^2.


MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]
q[n_]:=SQ[Floor[Sqrt[n]]]
a[n_]:=Sum[If[q[nPrime[k]], 1, 0], {k, 1, PrimePi[n]}]
Table[a[n], {n, 1, 70}]


CROSSREFS

Cf. A000040, A000290, A237705, A237706, A237710, A237720.
Sequence in context: A259977 A341434 A115312 * A254296 A248371 A237768
Adjacent sequences: A237718 A237719 A237720 * A237722 A237723 A237724


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 12 2014


STATUS

approved



