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A332086
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a(n) = pi(prime(n) + n) - n, where pi is the prime counting function.
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6
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1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 4, 4, 4, 4, 3, 4, 4, 6, 5, 5, 4, 4, 4, 4, 6, 6, 6, 6, 7, 6, 7, 8, 7, 7, 6, 6, 8, 7, 8, 7, 8, 10, 9, 9, 10, 9, 9, 8, 9, 10, 9, 8, 8, 8, 7, 9, 10, 10, 9, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13
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OFFSET
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1,6
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COMMENTS
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This sequence is related to a theorem of Lu and Deng (see LINKS): “The prime gap of a prime number is less than or equal to the prime count of the prime number”, which is equivalent to “There exists at least one prime number between p and p+pi(p)+1”, or pi(p+pi(p)) - pi(p) > 1, where pi is prime counting function. The n-th term of the sequence, a(n), is the number of prime number between the n-th prime number p_n and p_n + pi(p_n) + 1. According to the theorem, a(n) >= 1.
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LINKS
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FORMULA
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a(n) = pi(prime(n) + n) - n.
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EXAMPLE
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a(1) = pi(p_1 + 1) - 1 = pi(2 + 1) - 1 = 2 - 1 = 1;
a(2) = pi(p_2 + 2) - 2 = pi(3 + 2) - 2 = 3 - 2 = 1;
a(6) = pi(p_6 + 6) - 6 = pi(13 + 6) - 6 = 8 - 6 = 2;
a(80) = pi(p_80 + 80) - 80 = pi(409 + 80) - 80 = 93 - 80 = 13.
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MAPLE
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f:= n -> numtheory:-pi(ithprime(n)+n)-n:
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MATHEMATICA
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a[n_] := PrimePi[Prime[n] + n] - n; Array[a, 100] (* Amiram Eldar, Aug 23 2020 *)
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PROG
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(Python)
from sympy import prime, primepi
for n in range(1, 1001):
a = primepi(prime(n) + n) - n
print(a)
(PARI) a(n) = primepi(prime(n) + n) - n; \\ Michel Marcus, Aug 23 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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