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0, 0, 0, 1, 1, 3, 3, 3, 4, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 12, 14, 14, 14, 15, 17, 17, 17, 17, 17, 17, 17, 19, 21, 23, 23, 23, 25, 27, 27, 27, 27, 27, 27, 27, 29, 29, 29, 30, 30, 32, 32, 32, 32, 34, 34, 36, 38, 38, 38, 38, 40, 40, 40, 42, 42, 42, 42, 44, 44, 44, 44, 44, 46
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,6
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COMMENTS
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Sum of the Dirichlet convolution of the characteristic function of primes (A010051) with itself from 1 to n.
(a(n) + A000720(floor(sqrt(n))))/2 equals the number of semiprimes <= n.
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LINKS
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FORMULA
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a(n) = 2*Sum_{p prime <= sqrt(n)} A000720(floor(n/p)) - A000720(floor(sqrt(n)))^2.
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MAPLE
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a:= proc(n) option remember; `if`(n<4, 0, a(n-1) +
`if`(numtheory[bigomega](n)=2, `if`(issqr(n), 1, 2), 0))
end:
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MATHEMATICA
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f[n_] := DivisorSum[n, 1 &, PrimeQ[#] && PrimeQ[n/#] &]; Accumulate @ Array[f, 100] (* Amiram Eldar, May 20 2020 *)
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PROG
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(PARI) a(n) = my(s=sqrtint(n)); 2*sum(k=1, s, if(isprime(k), primepi(n\k), 0)) - primepi(s)^2;
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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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