



2, 2, 5, 3, 2, 2, 14, 4, 3, 3, 4, 1, 4, 3, 45, 3, 6, 6, 6, 5, 3, 6, 4, 5, 5, 6, 3, 5, 4, 6, 140, 12, 5, 9, 8, 11, 8, 5, 8, 8, 12, 8, 9, 7, 7, 8, 7, 6, 7, 9, 10, 5, 8, 11, 9, 8, 8, 7, 7, 9, 9, 7, 471, 14, 12, 15, 17, 15, 14, 13, 15, 14, 17, 12, 16, 16, 9, 17, 14, 12
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OFFSET

1,1


COMMENTS

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 12.
The terms of A020330 are usually called "binary squares". Our conjecture is an analog of Legendre's conjecture that for each n = 1,2,3,... there is a prime between n^2 and (n+1)^2.
Those a(2^n1) = pi(2*4^n+2^n)  pi(4^n) are relatively large, where pi(x) is the primecounting function given by A000720.
We have verified that a(n) > 0 for all n = 1..2*10^7.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
Wikipedia, Legendre's conjecture


EXAMPLE

a(1) = 2 since 5 and 7 are the only primes in the interval (A020330(1), A020330(2)) = (3, 10).
a(12) = 1 since 211 is the only prime greater than A020330(12) = 204 and smaller than A020330(13) = 221.
a(8191) = a(2^13  1) = pi(2^27 + 2^13)  pi(2^26) = 3646196.


MATHEMATICA

f[n_]:=f[n]=(2^(Floor[Log[2, n]]+1)+1)*n;
a[n_]:=a[n]=PrimePi[f[n+1]1]PrimePi[f[n]];
Table[a[n], {n, 1, 80}]


CROSSREFS

Cf. A000040, A000720, A014085, A020330.
Sequence in context: A132850 A076561 A132851 * A146316 A258803 A157495
Adjacent sequences: A293830 A293831 A293832 * A293834 A293835 A293836


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 16 2017


STATUS

approved



