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A015614
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a(n) = -1 + Sum_{i=1..n} phi(i).
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26
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0, 1, 3, 5, 9, 11, 17, 21, 27, 31, 41, 45, 57, 63, 71, 79, 95, 101, 119, 127, 139, 149, 171, 179, 199, 211, 229, 241, 269, 277, 307, 323, 343, 359, 383, 395, 431, 449, 473, 489, 529, 541, 583, 603, 627, 649, 695, 711, 753, 773, 805, 829, 881, 899, 939, 963
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OFFSET
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1,3
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COMMENTS
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Number of elements in the set {(x,y): 1 <= x < y <= n, 1=gcd(x,y)}. - Michael Somos, Jun 13 1999
Number of fractions in (Haros)-Farey series of order n.
The asymptotic limit for the sequence is a(n) ~ 3*n^2/Pi^2. - Martin Renner, Dec 12 2011
2*a(n) is the number of proper fractions reduced to lowest terms with numerator and denominator less than or equal to n in absolute value. - Stefano Spezia, Aug 09 2019
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REFERENCES
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Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, pp. 170-171.
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LINKS
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FORMULA
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G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} mu(k) * x^k / (1 - x^k)^2). - Ilya Gutkovskiy, Feb 14 2020
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EXAMPLE
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x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 11*x^6 + 17*x^7 + 21*x^8 +27*x^9 + ...
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MAPLE
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with(numtheory): a:=n->add(phi(i), i=1..n): seq(a(n)-1, n=1..60); # Muniru A Asiru, Jul 31 2018
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MATHEMATICA
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Table[Sum[EulerPhi[m], {m, 1, n}]-1, {n, 1, 56}] (* Geoffrey Critzer, May 16 2014 *)
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PROG
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(Haskell)
(PARI) {a(n) = if( n<1, 0, sum(k=1, n, eulerphi(k), -1))} /* Michael Somos, Sep 06 2013 */
(GAP) List([1..60], n->Sum([1..n], i->Phi(i)))-1; # Muniru A Asiru, Jul 31 2018
(Magma) [-1+&+[EulerPhi(i): i in [1..n]]:n in [1..56]]; // Marius A. Burtea, Aug 09 2019
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
if n == 0:
return -1
c, j = 2, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
j, k1 = j2, n//j2
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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