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A216265
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Number of primes between n^3 - n and n^3.
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3
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0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 0, 2, 3, 2, 2, 2, 2, 1, 2, 3, 4, 1, 3, 3, 2, 3, 3, 3, 2, 1, 3, 2, 4, 4, 3, 2, 1, 2, 7, 4, 2, 2, 4, 3, 4, 7, 3, 5, 7, 4, 6, 5, 4, 2, 8, 4, 3, 4, 2, 5, 7, 7, 4, 3, 8, 4, 1, 3, 2, 10, 4, 5, 4, 6, 7, 8, 6, 6, 1, 6, 8, 8, 7, 7, 6, 7, 4, 10
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OFFSET
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1,10
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COMMENTS
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Conjecture: a(n) > 0 for n > 13.
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LINKS
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FORMULA
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EXAMPLE
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a(9) = 1 because between 9^3 - 9 and 9^3 there is just one prime (727).
a(10) = 2 because between 10^3 - 10 and 10^3 there are two primes (991 and 997).
a(11) = 2 because between 11^3 - 11 and 11^3 there are two primes (1321 and 1327).
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MAPLE
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a:= n-> add(`if`(isprime(t), 1, 0), t=n^3-n..n^3):
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MATHEMATICA
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Table[PrimePi[n^3] - PrimePi[n^3 - n], {n, 100}] (* Alonso del Arte, Mar 17 2013 *)
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PROG
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(Java)
import java.math.BigInteger;
public static void main (String[] args) {
for (long n = 1; n < (1 << 21); n++) {
long cube = n*n*n, c = 0;
for (long k = cube - n; k < cube; ++k) {
BigInteger b1 = BigInteger.valueOf(k);
if (b1.isProbablePrime(2)) {
if (b1.isProbablePrime(80))
++c;
}
}
System.out.printf("%d, ", c);
}
}
} // Ratushnyak
(PARI)
default(primelimit, 10^7);
a(n) = primepi(n^3) - primepi(n^3-n);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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