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A293833 Number of primes p with A020330(n) < p < A020330(n+1). 1
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 (list; graph; refs; listen; history; text; internal format)
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^n-1) = pi(2*4^n+2^n) - pi(4^n) are relatively large, where pi(x) is the prime-counting function given by A000720.

We have verified that a(n) > 0 for all n = 1..2*10^7.

LINKS

Zhi-Wei 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

Zhi-Wei Sun, Oct 16 2017

STATUS

approved

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Last modified March 30 21:56 EDT 2020. Contains 333132 sequences. (Running on oeis4.)