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A064608
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Partial sums of A034444: sum of number of unitary divisors from 1 to n.
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15
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1, 3, 5, 7, 9, 13, 15, 17, 19, 23, 25, 29, 31, 35, 39, 41, 43, 47, 49, 53, 57, 61, 63, 67, 69, 73, 75, 79, 81, 89, 91, 93, 97, 101, 105, 109, 111, 115, 119, 123, 125, 133, 135, 139, 143, 147, 149, 153, 155, 159, 163, 167, 169, 173, 177, 181, 185, 189, 191, 199, 201
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OFFSET
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1,2
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COMMENTS
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a(n) = Sum_{k<=n} 2^omega(k) where omega(k) is the number of distinct primes in the factorization of k. - Benoit Cloitre, Apr 16 2002
a(n) is the number of (p, q) lattice points that are visible from (0, 0), where p and q satisfy: p >= 1, q >= 1, p * q <= n. - Luc Rousseau, Jul 09 2017
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REFERENCES
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E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig 1909 (Chelsea reprint 1953), p. 594.
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LINKS
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E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
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FORMULA
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a(n) = n*log(n)/zeta(2) + O(n) where zeta(2) = Pi^2/6. - Benoit Cloitre, Apr 16 2002
Mertens's theorem (1874): a(n) = Sum_{k<=n} ud(k) = (n/Zeta(2))*(log(n) + 2*gamma - 1 - 2*Zeta'(2)/Zeta(2)) + O(sqrt(n)*log(n)), where gamma is the Euler-Mascheroni constant A001620. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
G.f.: (1/(1 - x))*Sum_{k>=1} mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
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MAPLE
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MATHEMATICA
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Accumulate@ Array[2^PrimeNu[#] &, {61}] (* Amiram Eldar, Oct 21 2019 *)
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PROG
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(PARI) { for (n=1, 80, a=sum(k=1, n, moebius(k)^2*floor(n/k)); write("b064608.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009
(PARI) a(n)=sum(k=1, sqrtint(n), moebius(k)*(2*sum(l=1, sqrtint(n\(k*k)), n\(k*k*l))-sqrtint(n\(k*k))^2)); \\ More efficient formula for large n values (up to 10^14)
(Python)
from sympy.ntheory.factor_ import primenu
def A064608(n): return sum(1<<primenu(m) for m in range(1, n+1)) # Chai Wah Wu, Sep 07 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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