A064608 Partial sums of A034444: sum of number of unitary divisors from 1 to n.

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

Offset

1,2

Comments

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

References

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig 1909 (Chelsea reprint 1953), p. 594.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019.

E. Cohen, The number of unitary divisors of an integer, Am. Math. Mon. 67, 879-880 (1960).

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

F. Mertens, Uber einige asymptotische Gesetze der Zahlentheorie, J. Reine Angew. Math., 77 (1874), 289-338.

V. Sitaramaiah and M.V. Subbarao, Unitary divisor problem for arithmetic progressions, Annales Univ. Sci. Budapest., Sect. Comp. 32 (2010) 73-89.

D. Suryanarayana and V. Siva Rama Prasad, The number of k-free divisors of an integer, Acta Arithmetica XVII (1971), 345-354.

D. Zhang and W. Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq. 13 (2010), 10.4.7. Eq (8) for asymptotics.

Formula

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).

a(n) = n*log(n)/zeta(2) + O(n) where zeta(2) = Pi^2/6. - Benoit Cloitre, Apr 16 2002

a(n) = Sum_{k=1..n} mu(k)^2*floor(n/k). - 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

Maple

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

Mathematica

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 *)
Accumulate@ Array[2^PrimeNu[#] &, {61}] (* Amiram Eldar, Oct 21 2019 *)

Prog

(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)
vector(80, i, a(i)) \\ Jerome Raulin, Nov 01 2015
(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

Crossrefs

Cf. A001620, A001221, A034444, A064610, A069212.

Sequence in context: A063425 A215806 A333369 * A024893 A337810 A355773

Adjacent sequences: A064605 A064606 A064607 * A064609 A064610 A064611

Keyword

nonn,easy

Author

Labos Elemer, Sep 24 2001

Status

approved