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%I #82 Sep 08 2023 01:29:44
%S 1,3,5,7,9,13,15,17,19,23,25,29,31,35,39,41,43,47,49,53,57,61,63,67,
%T 69,73,75,79,81,89,91,93,97,101,105,109,111,115,119,123,125,133,135,
%U 139,143,147,149,153,155,159,163,167,169,173,177,181,185,189,191,199,201
%N Partial sums of A034444: sum of number of unitary divisors from 1 to n.
%C 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
%C 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
%D E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig 1909 (Chelsea reprint 1953), p. 594.
%H Amiram Eldar, <a href="/A064608/b064608.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%H Masum Billal, <a href="https://arxiv.org/abs/1909.07823">Number of Ways To Write as Product of Co-prime Numbers</a>, arXiv:1909.07823 [math.GM], 2019.
%H E. Cohen, <a href="http://www.jstor.org/stable/2309455">The number of unitary divisors of an integer</a>, Am. Math. Mon. 67, 879-880 (1960).
%H E. Landau, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ABV2766.0002.001">Handbuch der Lehre von der Verteilung der Primzahlen</a>, vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
%H E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, <a href="http://name.umdl.umich.edu/ABV2766.0001.001">vol. 1</a> and <a href="http://name.umdl.umich.edu/ABV2766.0002.001">vol. 2</a>, Leipzig, Berlin, B. G. Teubner, 1909.
%H F. Mertens, <a href="http://dx.doi.org/10.1515/crll.1874.77.289">Uber einige asymptotische Gesetze der Zahlentheorie</a>, J. Reine Angew. Math., 77 (1874), 289-338.
%H V. Sitaramaiah and M.V. Subbarao, <a href="http://ac.inf.elte.hu/Vol_032_2010/073.pdf">Unitary divisor problem for arithmetic progressions</a>, Annales Univ. Sci. Budapest., Sect. Comp. 32 (2010) 73-89.
%H D. Suryanarayana and V. Siva Rama Prasad, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa17/aa1743.pdf">The number of k-free divisors of an integer</a>, Acta Arithmetica XVII (1971), 345-354.
%H D. Zhang and W. Zhai, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Zhang/zhang10.html">Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n</a>, J. Int. Seq. 13 (2010), 10.4.7. Eq (8) for asymptotics.
%F a(n) = a(n-1) + A034444(n) = a(n-1) + 2^A001221(n) Sum_{j=1..n} ud(j) where ud(j) = A034444(j) = 2^A001221(n).
%F a(n) = n*log(n)/zeta(2) + O(n) where zeta(2) = Pi^2/6. - _Benoit Cloitre_, Apr 16 2002
%F a(n) = Sum_{k=1..n} mu(k)^2*floor(n/k). - _Benoit Cloitre_, Apr 16 2002
%F 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
%F G.f.: (1/(1 - x))*Sum_{k>=1} mu(k)^2*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 03 2017
%p with(numtheory): A064608:=n->add(mobius(k)^2*floor(n/k), k=1..n): seq(A064608(n), n=1..100); # _Wesley Ivan Hurt_, Dec 05 2015
%t a[n_] := Count[Divisors@ n, d_ /; GCD[d, n/d] == 1]; Accumulate@ Array[a, {61}] (* _Michael De Vlieger_, Oct 21 2015, after _Jean-François Alcover_ at A034444 *)
%t Accumulate@ Array[2^PrimeNu[#] &, {61}] (* _Amiram Eldar_, Oct 21 2019 *)
%o (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
%o (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)
%o vector(80,i,a(i)) \\ _Jerome Raulin_, Nov 01 2015
%o (Python)
%o from sympy.ntheory.factor_ import primenu
%o def A064608(n): return sum(1<<primenu(m) for m in range(1,n+1)) # _Chai Wah Wu_, Sep 07 2023
%Y Cf. A001620, A001221, A034444, A064610, A069212.
%K nonn,easy
%O 1,2
%A _Labos Elemer_, Sep 24 2001
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