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1, 1, 1, 3, 3, 5, 5, 6, 8, 9, 9, 11, 11, 12, 13, 14, 14, 16, 16, 17, 19, 20, 20, 21, 22, 24, 25, 27, 27, 28, 28, 29, 30, 32, 33, 34, 34, 35, 36, 37, 37, 39, 39, 40, 42, 43, 43, 44, 45, 46, 47, 49, 49, 50, 51, 52, 54, 55, 55, 57, 57, 58, 59, 60, 62, 63, 63, 64
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OFFSET
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1,4
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COMMENTS
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It can be shown that there is at least one prime number between n-pi(n) and n for n >= 3, or pi(n-1)-pi(n-pi(n)) >= 1. Since a(n)=n-pi(n)+pi(n-pi(n)) <= n-pi(n-1)+pi(n-pi(n)) <= n-1, we have a(n) < n for n > 1.
a(n)-a(n-1) = 1 - (pi(n)-pi(n-1)) + pi(n-pi(n)) - pi(n-(1+pi(n-1))), where pi(n)-pi(n-1) <= 1 and 1+pi(n-1) >= pi(n) or pi(n-(1+pi(n-1))) <= pi(n-pi(n)). Thus, a(n) - a(n-1) >= 0, meaning that this is a nondecreasing sequence.
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LINKS
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FORMULA
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a(n) = n - pi(n) + pi(n - pi(n)), where pi(n) is the prime count of n.
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MATHEMATICA
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PROG
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(Python)
from sympy import primepi
for n in range(1, 10001):
b = n - primepi(n)
a = b + primepi(b)
print(a)
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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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