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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A002088 Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.

Original entry on oeis.org

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Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Number of elements in the set {(x,y): 1 <= x <= y <= n, 1=gcd(x,y)}. - Michael Somos, Jun 13 1999
Sum_{k=1..n} phi(k) gives the number of distinct arithmetic progressions which contain an infinite number of primes and whose difference does not exceed n. E.g., {1k+1}, {2k+1}, {3k+1, 3k+2}, {4k+1, 4k+3}, {5k+1, ..5k+4} means 10 sequences. - Labos Elemer, May 02 2001
The quotient A024916(n)/a(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 = zeta(2)^2 = A098198 ~2.705808084277845. - Labos Elemer, Sep 20 2004 (corrected by Peter Pein, Apr 28 2009)
Also the number of rationals p/q in (0,1] with denominators q<=n. - Franz Vrabec, Jan 29 2005
a(n) is the number of initial segments of Beatty sequences for real numbers > 1, cut off when the next term in the sequence would be >= n. For example, the sequence 1,2 is included for n=3 and n=4, but not for n >= 5 because the next term of the Beatty sequence must be 3 or 4. Problem suggested by David W. Wilson. - Franklin T. Adams-Watters, Oct 19 2006
Number of complex numbers satisfying any one of {x^1=1, x^2=1, x^3=1, x^4=1, x^5=1, ..., x^n=1}. - Paul Smith (math.idiot(AT)gmail.com), Mar 19 2007
a(n+2) equals the number of Sturmian words of length n which are 'special', prefix of two Sturmian words of length n+1. - Fred Lunnon, Sep 05 2010
For n > 1: A020652(a(n)) = 1 and A038567(a(n)) = n; for n > 0: A214803(a(n)) = 1. - Reinhard Zumkeller, Jul 29 2012
Also number of elements in the set {(x,y): 1 <= x + y <= n, x >= 0, y > 0, with x and y relatively prime integers}. Thus, the number of reduced rational numbers x/y with x nonnegative, y positive, and x + y <= n. (For n >= 1, 0 <= x/y <= n - 1, clearly including each integer in this interval.) - Rick L. Shepherd, Apr 08 2014
This function, the partial sums of phi = A000010, is sometimes denoted by (uppercase) Phi. - M. F. Hasler, Apr 18 2015
From Roger Ford, Jan 16 2016: (Start)
For n >= 1: a(n) is the number of perfect arched semi-meander solutions with n arches. To be perfect the number of arch groupings must equal the number of arches with a length of 1 in the current generation and every preceding generation.
Example: p is the number of arches with length 1 (/\), g is the number of arch groups (-), n is number of arches in the top half of a semi-meander solution
/\
/\ //\\
//\\-/\-///\\\- n=6 p=3 g=3 Each preceding arch configuration
/\ /\ is formed by attaching the arch
/\-//\\-//\\- n=5 p=3 g=3 end in the first position and the
/\ arch end in the last position.
//\\
///\\\-/\- n=4 p=2 g=2
/\
//\\-/\- n=3 p=2 g=2
/\-/\- n=2 p=2 g=2
/\- n=1 p=1 g=1. (End)
a(n) is the number of distinct lists of binary words of length n that are balanced (Sturmian). - Dan Rockwell, Will Wodrich, Aaliyah Fiala, and Bob Burton, May 30 2019
2013 IMO Problem 6 shows that a(n) is the number of ways to arrange the numbers 0, 1, ..., n on a circle such that for any numbers 0 <= a < b < c < d <= n, the chord joining a and d does not intersect with the chord intersecting b and c, with rotation counted as same. - Yifan Xie, Aug 26 2025

Examples

			G.f. = x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 12*x^6 + 18*x^7 + 22*x^8 + 28*x^9 + ...

References

  • A. Beiler, Recreations in the Theory of Numbers, Dover Publications, 1966, Chap. XVI.
  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 138.
  • M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972, p. 6.
  • D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section I.21.
  • I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 94, Problem 11.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 111.

Links

Crossrefs

Cf. A000010, A015614, A005728, A067282, A001088, A134542, A002321.

Programs

  • GAP
    List([1..60],n->Sum([1..n],i->Phi(i))); # Muniru A Asiru, Jul 31 2018
  • Haskell
    a002088 n = a002088_list !! n
a002088_list = scanl (+) 0 a000010_list -- Reinhard Zumkeller, Jul 29 2012
  • Magma
    [&+[EulerPhi(i): i in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Aug 01 2018
  • Maple
    with(numtheory): A002088:=n->add(phi(i),i=1..n): seq(A002088(n), n=0..70);
  • Mathematica
    Table[Plus @@ EulerPhi[Range[n]], {n, 0, 57}] (* Alonso del Arte, May 30 2006 *)
Accumulate[EulerPhi[Range[0,60]]] (* Harvey P. Dale, Aug 27 2011 *)
  • PARI
    a(n)=sum(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Jun 16 2011
  • PARI
    a(n)=my(s=1); forsquarefree(k=1,n,s+=(n\k[1])^2*moebius(k)); s/2 \\ Charles R Greathouse IV, Oct 15 2021
  • PARI
    first(n)=my(v=vector(n),s); forfactored(k=1,n, v[k[1]]=s+=eulerphi(k)); v \\ Charles R Greathouse IV, Oct 15 2021
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A002088(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(2*A002088(k1)-1)
        j, k1 = j2, n//j2
    return (n*(n-1)-c+j)//2 # Chai Wah Wu, Mar 24 2021
  • Sage
    [sum(euler_phi(k) for k in (1..n)) for n in (0..60)] # G. C. Greubel, Nov 25 2018

Formula

a(n) = (3*n^2)/(Pi^2) + O(n log n).
More precisely, a(n) = (3/Pi^2)*n^2 + O(n*(log(n))^(2/3)*(log(log(n)))^(4/3)), (A. Walfisz 1963). - Benoit Cloitre, Feb 02 2003
a(n) = (1/2)*Sum_{k>=1} mu(k)*floor(n/k)*floor(1+n/k). - Benoit Cloitre, Apr 11 2003
a(n) = A000217(n) - A063985(n) = A018805(n) - A015614(n). - Reinhard Zumkeller, Jan 21 2013
A slightly simpler version of Cloitre's formula is a(n) = 1/2 + Sum_{k=1..oo} floor(n/k)^2*mu(k)/2. - Bill Gosper, Jul 25 2020
The quotient A024916(n)/a(n) = SummatorySigma/SummatoryTotient as n increases seems to approach (Pi^4)/36 = Zeta(2)^2 = 2.705808084277845. See also A067282. - Labos Elemer, Sep 21 2004
A024916(n)/a(n) = zeta(2)^2 + O(log(n)/n). This follows from asymptotic formulas for the sequences. - Franklin T. Adams-Watters, Oct 19 2006
Row sums of triangle A134542. - Gary W. Adamson, Oct 31 2007
G.f.: (Sum_{n>=1} mu(n)*x^n/(1-x^n)^2)/(1-x), where mu(n) = A008683(n). - Mamuka Jibladze, Apr 06 2015
a(n) = A005728(n) - 1, for n >= 0. - Wolfdieter Lang, Nov 22 2016
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (M(floor(n/k)) - M(floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k) * floor(n/k) * floor(1+n/k))/2, where M(k) is the Mertens function (A002321) and mu(k) is the Moebius function (A008683). - Daniel Suteu, Nov 23 2018
a(n) = A015614(n)+1. - R. J. Mathar, Apr 26 2023
a(n) = A000217(n) - Sum{k=2..n} a(floor(n/k)). From summing over Id = 1 (Dirichlet convolution) phi. - Jason Xu, Jul 31 2024
a(n) = Sum_{k=1..n} k*A002321(floor(n/k)). - Ridouane Oudra, Jul 03 2025

Extensions

Additional comments from Len Smiley

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