A002088 -id:A002088 - cp's OEIS frontend

A038567 Denominators in canonical bijection from positive integers to positive rationals <= 1.

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16

Offset: 0

N. J. A. Sloane

n occurs phi(n) times (cf. A000010).
Least k such that phi(1) + phi(2) + phi(3) + ... + phi(k) >= n. - Benoit Cloitre, Sep 17 2002
Sum of numerator and denominator of fractions arranged by Cantor's ordering (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, ...) with equivalent fractions removed. - Ron R. King, Mar 07 2009 [This applies to a(1, 2, ...) without initial term a(0) = 1 which could correspond to 0/1. - Editor's Note.]
Care has to be taken in considering the offset which may be 0 or 1 in related sequences (see crossrefs), e.g., A038568 & A038569 also have offset 0, in A038566 offset has been changed to 1. - M. F. Hasler, Oct 18 2021

			Arrange fractions by increasing denominator then by increasing numerator: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ...: this is A038566/A038567.

S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.
Hans Lauwerier, Fractals, Princeton University Press, 1991, p. 23.

David Wasserman, Table of n, a(n) for n = 0..100000
Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633. [Wayback Machine link]
Index entries for "core" sequences.
Index entries for sequences related to enumerating the rationals.
Index entries for sequences related to Stern's sequences.

Cf. A020652, A020653, A038566 - A038569, A093602, A182972, A182973 - A182976.
A054427 gives mapping to Stern-Brocot tree.
Cf. A037162.

import Data.List (genericTake)
a038567 n = a038567_list !! n
a038567_list = concatMap (\x -> genericTake (a000010 x) $ repeat x) [1..]
-- Reinhard Zumkeller, Dec 16 2013, Jul 29 2012

with (numtheory): A038567 := proc (n) local sum, k; sum := 1: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)

a[n_] := (k = 0; While[ Total[ EulerPhi[ Range[k]]] <= n, k++]; k); Table[ a[n], {n, 0, 77}] (* Jean-François Alcover, Dec 08 2011, after Pari *)
Flatten[Table[Table[n,{EulerPhi[n]}],{n,20}]] (* Harvey P. Dale, Mar 12 2013 *)

a(n)=if(n<0,0,s=1; while(sum(i=1,s,eulerphi(i))

from sympy import totient
def a(n):
    s=1
    while sum(totient(i) for i in range(1, s + 1))Indranil Ghosh, May 23 2017

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)*((A002088(k1)<<1)-1)
        j, k1 = j2, n//j2
    return n*(n-1)-c+j>>1
def A038567(n):
    kmin, kmax = 0, 1
    while A002088(kmax) <= n:
        kmax <<= 1
    kmin = kmax>>1
    while True:
        kmid = kmax+kmin>>1
        if A002088(kmid) > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Jun 10 2025

From Henry Bottomley, Dec 18 2000: (Start)
a(n) = A020652(n) + A020653(n) for all n > 0, e.g., a(1) = 2 = 1 + 1 = A020652(1) + A020653(1). [Corrected and edited by M. F. Hasler, Dec 10 2021]
n = a(A015614(n)) = a(A002088(n)) - 1 = a(A002088(n-1)). (End)
a(n) = A002024(A169581(n)). - Reinhard Zumkeller, Dec 02 2009
a(A002088(n)) = n for n > 1. - Reinhard Zumkeller, Jul 29 2012
a(n) = A071912(2*n+1). - Reinhard Zumkeller, Dec 16 2013
a(n) ~ c * sqrt(n), where c = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Dec 27 2024

More terms from Erich Friedman

A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

Original entry on oeis.org

1, 1, 1, 2, 4, 16, 32, 192, 768, 4608, 18432, 184320, 737280, 8847360, 53084160, 424673280, 3397386240, 54358179840, 326149079040, 5870683422720, 46965467381760, 563585608581120, 5635856085811200, 123988833887846400, 991910671102771200, 19838213422055424000

Offset: 0

Simon Plouffe

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j) for 1 <= i,j <= n [Smith and Mansion]. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001

The matrix M(i,j) = gcd(i,j) is sequence A003989. - Michael Somos, Jun 25 2012

			a(2) = 1 because the matrix M is: [1,1; 1,2] and det(A) = 1.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 598.
M. Petkovsek et al., A=B, Peters, 1996, p. 21.

Antoine Mathys, Table of n, a(n) for n = 0..496 (first 100 terms by T. D. Noe)
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
E. C. Catalan, Théorème de MM. Smith et Mansion, Nouvelle correspondance mathématique, 4 (1878) 103-112. [_Philippe Deléham_, Dec 22 2003]
Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, 76 (2003), 392-394.
P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 81-82.
Mathoverflow, Asymptotics of product of Euler's totient function, 2016.
H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.
Eric Weisstein's World of Mathematics, Le Paige's Theorem
Index to divisibility sequences

Cf. A000010, A060238, A060239, A059381, A059382, A059383, A059384, A002088.

Cf. A003989.

List([1..30],n->Product([1..n],i->Phi(i))); # Muniru A Asiru, Jul 31 2018

a001088 n = a001088_list !! (n-1)
a001088_list = scanl1 (*) a000010_list
-- Reinhard Zumkeller, Mar 04 2012

with(numtheory,phi); A001088 := proc(n) local i; mul(phi(i),i=1..n); end;
seq(A001088(n), n=0..30);

A001088[n_]:=Times@@EulerPhi/@Range[n]; Table[A001088[n], {n, 30}] (* Enrique Pérez Herrero, Sep 19 2010 *)
Rest[FoldList[Times,1,EulerPhi[Range[30]]]] (* Harvey P. Dale, Dec 09 2011 *)

a(n)=prod(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Mar 04 2012

a(n) = phi(1) * phi(2) * ... * phi(n).

Limit_{n->infinity} a(n)^(1/n) / n = exp(-1) * A124175 = 0.205963050288186353879675428232497466485878059342058515016427881513657493... (see Mathoverflow link). - Vaclav Kotesovec, Jun 09 2021

a(0)=1 prepended by Alois P. Heinz, Jul 19 2023

A020652 Numerators in canonical bijection from positive integers to positive rationals.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 3, 4, 1, 5, 1, 2, 3, 4, 5, 6, 1, 3, 5, 7, 1, 2, 4, 5, 7, 8, 1, 3, 7, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 5, 7, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 3, 5, 9, 11, 13, 1, 2, 4, 7, 8, 11, 13, 14, 1, 3, 5, 7, 9, 11, 13, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 5

Offset: 1

David W. Wilson

a(A002088(n)) = 1 for n > 1. - Reinhard Zumkeller, Jul 29 2012
When read as an irregular table with each 1 entry starting a new row, then the n-th row consists of the set of multiplicative units of Z_{n+1}. These rows form a group under multiplication mod n. - Tom Edgar, Aug 20 2013
The pair of sequences A020652/A020653 is defined by ordering the positive fractions p/q (reduced to lowest terms) by increasing p+q, then increasing p: 1/1; 1/2, 2/1; 1/3, 3/1; 1/4, 2/3, 3/2, 4/1; 1/5, 5/1; 2/5, 3/4, 4/3, 5/2; etc. For given p+q, there are A000010(p+q) fractions, listed starting at index A002088(p+q-1). - M. F. Hasler, Mar 06 2020

			Arrange positive fractions < 1 by increasing denominator then by increasing numerator: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6 ... (this is A020652/A038567). - _William Rex Marshall_, Dec 16 2010

S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.
Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.
H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.

David Wasserman, Table of n, a(n) for n = 1..100000
Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.
Index entries for "core" sequences
Index entries for sequences related to enumerating the rationals
Index entries for sequences related to Stern's sequences

Essentially the same as A038566, which is the main entry for this sequence.
Cf. A020653, A038567-A038569, A182972-A182976.
A054424 gives mapping to Stern-Brocot tree.
Cf. A037161.

a020652 n = a020652_list !! (n-1)
a020652_list = map fst [(u,v) | v <- [1..], u <- [1..v-1], gcd u v == 1]
-- Reinhard Zumkeller, Jul 29 2012

with (numtheory): A020652 := proc (n) local sum, j, k; sum := 0: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: sum := sum - phi(k-1): j := 1; while sum < n do: if gcd(j,k-1) = 1 then sum := sum + 1: fi: j := j+1: od: RETURN (j-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com), Nov 06 2001

Reap[Do[If[GCD[num, den] == 1, Sow[num]], {den, 1, 20}, {num, 1, den-1}] ][[2, 1]] (* Jean-François Alcover, Oct 22 2012 *)

a(n)=my(s,j=1,k=1);while(sCharles R Greathouse IV, Feb 07 2013

from sympy import totient, gcd
def a(n):
    s=0
    k=2
    while sIndranil Ghosh, May 23 2017, after Ulrich Schimke's MAPLE code

A104141 Decimal expansion of 3/Pi^2.

Original entry on oeis.org

3, 0, 3, 9, 6, 3, 5, 5, 0, 9, 2, 7, 0, 1, 3, 3, 1, 4, 3, 3, 1, 6, 3, 8, 3, 8, 9, 6, 2, 9, 1, 8, 2, 9, 1, 6, 7, 1, 3, 0, 7, 6, 3, 2, 4, 0, 1, 6, 7, 3, 9, 6, 4, 6, 5, 3, 6, 8, 2, 7, 0, 9, 5, 6, 8, 2, 5, 1, 9, 3, 6, 2, 8, 8, 6, 7, 0, 6, 3, 2, 3, 5, 7, 3, 6, 2, 7, 8, 2, 1, 7, 7, 6, 8, 6, 5, 5, 1, 2, 8

Offset: 0

Lekraj Beedassy, Mar 07 2005

3/Pi^2 is the limit of (Sum_{k=1..n} phi(k))/n^2, where phi(k) is Euler's totient A000010(k), i.e., of A002088(n)/A000290(n) as n tends to infinity.
The previous comment in the context of Farey series means that the length of the n-th Farey series can be approximated by multiplying this constant by n^2, "and that the approximation gets proportionally better as n gets larger", according to Conway and Guy. - Alonso del Arte, May 28 2011
The asymptotic density of the sequences of squarefree numbers with even number of prime factors (A030229), odd number of prime factors (A030059), and coprime to 6 (A276378). - Amiram Eldar, May 22 2020

			3/Pi^2 = 0.303963550927013314331638389629...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 156.
L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 184.

Eric Weisstein's World of Mathematics, Farey Sequence
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
Index entries for transcendental numbers.

Cf. A000010, A002088, A000290.
Cf. A046642, A030229, A030059, A039956, A276378.
Cf. A013661, A195055, A306633, A082020, A088246.

l = RealDigits[N[3/Pi^2, 100]]; Prepend[First[l], Last[l]] (* Ryan Propper, Aug 04 2005 *)

3/Pi^2 \\ Charles R Greathouse IV, Mar 08 2013

Equals Sum_{n>=1} 1/A039956(n)^2. - Amiram Eldar, May 22 2020
From Terry D. Grant, Oct 31 2020: (Start)
Equals (-1)*zeta(0)/zeta(2).
Equals 1/(zeta(2)/2).
Equals 1/A195055.
Equals (1/2)*Sum_{k>=1} mu(k)/k^2. (End)
From Hugo Pfoertner, Apr 23 2024: (Start)
Equals A059956/2.
Equals A082020/5. (End)

More terms from Ryan Propper, Aug 04 2005

A015614 a(n) = -1 + Sum_{i=1..n} phi(i).

Original entry on oeis.org

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

Offset: 1

Olivier Gérard

nonn

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

			x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 11*x^6 + 17*x^7 + 21*x^8 +27*x^9 + ...

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, pp. 170-171.

Stanislav Sykora, Table of n, a(n) for n = 1..10000
James J. Sylvester, On the number of fractions contained in any Farey series of which the limiting number is given, in: London, Edinburgh and Dublin Philosophical Magazine (5th series) 15 (1883), p. 251.

Cf. A000010, A002088, A018805, A185670.

Column k=2 of triangle A186974.

List([1..60],n->Sum([1..n],i->Phi(i)))-1; # Muniru A Asiru, Jul 31 2018

a015614 = (subtract 1) . a002088  -- Reinhard Zumkeller, Jul 29 2012

[-1+&+[EulerPhi(i): i in [1..n]]:n in [1..56]]; // Marius A. Burtea, Aug 09 2019

with(numtheory): a:=n->add(phi(i),i=1..n): seq(a(n)-1,n=1..60); # Muniru A Asiru, Jul 31 2018

Table[Sum[EulerPhi[m],{m,1,n}]-1,{n,1,56}] (* Geoffrey Critzer, May 16 2014 *)
Table[Length[FareySequence[n]]-2,{n,60}] (* Harvey P. Dale, Jan 30 2025 *)

{a(n) = if( n<1, 0, sum(k=1,n,eulerphi(k), -1))} /* Michael Somos, Sep 06 2013 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A015614(n): # based on second formula in A018805
    if n == 0:
        return -1
    c, j = 2, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(2*A015614(k1)+1)
        j, k1 = j2, n//j2
    return (n*(n-1)-c+j)//2 # Chai Wah Wu, Mar 24 2021

a(n) = -1 + A002088(n).
a(n) = (A018805(n) - 1)/2. - Reinhard Zumkeller, Apr 08 2006
For n > 1: A214803(a(n)) = A165900(n-1). - Reinhard Zumkeller, Jul 29 2012
a(n) = A018805(n) - A002088(n). - Reinhard Zumkeller, Jan 21 2013
G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} mu(k) * x^k / (1 - x^k)^2). - Ilya Gutkovskiy, Feb 14 2020
a(n) = A000217(n-1) - A185670(n). - Hossein Mahmoodi, Jan 20 2022

More terms from Reinhard Zumkeller, Apr 08 2006

A048290 Numbers m that divide Sum_{k=1..m} phi(k).

Original entry on oeis.org

1, 2, 5, 6, 16, 25, 36, 249, 617, 1296, 13763, 76268, 189074, 783665, 1102394, 3258466, 3808854, 7971034, 15748051, 27746990, 41846733, 153673168, 195853251, 302167272, 402296412, 732683468, 807656448, 844492262, 848152352, 1122039882, 2258200198, 2438160726

Offset: 1

David J. Rusin

The odd terms of this sequence and A063986 are the same. - Jud McCranie, Jun 26 2005

			Euler sums are *1*, *2*, 4, 6, *10*, *12*, ..., *80*, ..., *510624*,... for n=1, 2, 3, 4, 5, 6, ..., 16, ...., 1296, ...

Donovan Johnson, Table of n, a(n) for n = 1..37 (terms < 10^12)
Edward A. Bender, Oren Patashnik and Howard Rumsey, Jr., Pizza Slicing, Phi's and the Riemann Hypothesis, American Mathematical Monthly, Vol. 101 (1994), pp. 307-317.
D. Rusin, Euler phi function

Cf. A000010, A002088. See A063986 for n divides Sum_{k=1..n} k-phi(k).

s = 0; Do[s = s + EulerPhi[n]; If[IntegerQ[s/n], Print[n]], {n, 1, 10^8}]

list(lim)=my(v=List(),s); for(k=1,lim, s+=eulerphi(k); if(s%k==0, listput(v, k))); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Sum_{k=1..m} phi(k) is about (3/Pi^2)*m^2 [cf. A002088, first formula].

Not obviously infinite; rough heuristics predict about 3/2 log(N) terms less than N, log(N) even ones and log(N)/2 odd ones.

10 more terms computed by Dean Hickerson
One more term from Robert G. Wilson v, Sep 07 2001
More terms from Naohiro Nomoto, Mar 22 2002
5 more terms from Jud McCranie, Jun 21 2005

A005598 a(n) = 1 + Sum_{i=1..n} (n-i+1)*phi(i).

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 36, 54, 76, 104, 136, 178, 224, 282, 346, 418, 498, 594, 696, 816, 944, 1084, 1234, 1406, 1586, 1786, 1998, 2228, 2470, 2740, 3018, 3326, 3650, 3994, 4354, 4738, 5134, 5566, 6016, 6490, 6980, 7510, 8052, 8636, 9240, 9868, 10518, 11214

Offset: 0

N. J. A. Sloane

Number of possible interleaving orders for n consecutive distinct values from two arithmetic progressions. ABABBBA is impossible, for example, because "ABA" implies that the spacing between B's must be greater than 1/2 the spacing between A's. But "ABBBA" implies that the B-spacing must be less than 1/2 the A-spacing. - Allan C. Wechsler, Mar 16 2005. Since the interchange of A's and B's gives essentially the same order pattern, all terms will be even for n>0.
The SemialgebraicComponents procedure in the Algebra`AlgebraicInequalities` package of Mathematica may be used to determine whether a particular pattern is possible. - John W. Layman, Mar 30 2005
Also, "digital lines": number of straight binary strings of length n [Dorst]. This was the original source for this sequence.
Also, the number of finite Sturmian words of length n. The considered orders are exactly the balanced words, which have been proved to be the factors of Sturmian sequences. An explicit formula was exhibited by Mignosi in 1991. Berstel and Pocchiola gave a geometric proof of this, using Euler's function for counting partitions of a unit cube. - Damien Jamet (jamet(AT)lirmm.fr), Apr 01 2005
The first difference of a(n) is the number of 'special' words, prefix of two Sturmian words of length n+1; see A002088. The second difference of a(n) is the number of palindromic 'bispecial' words, prefix and suffix of two Sturmian words of length n+1; see A000010. - Fred Lunnon, Sep 05 2010
Conjectured to be the number of regions in a Farey fan of order n. See A360042 for further details. - Scott R. Shannon, Jan 24 2023

			a(4) = 14 because of the 16 possible four-letter words from an alphabet of two letters, only AABB and BBAA are not possible interleaving orders for two arithmetic progressions.
For n=7, the pattern BAAAABA gives a possible ordering for the two arithmetic progressions {A, A+a, A+2a, A+3a,...} and {B, B+b, B+2b, B+3b,...} if the system of inequalities {a>0, b>0, A<B, B < A+a, A+4a<B+b, B+b < A+5a, A+5a<B+2b} has a solution. (Note: A<B is included to preclude a fifth A-term from lying between the two B-terms; similarly, A+5a<B+2b is included to preclude a second B-term from lying between the final two A-terms.) The SemialgebraicComponents procedure gives the solution {A,a,B,b}={0,1,1/8,4}; thus BAAAABA is one of the 54 possible orders of length 7. - _John W. Layman_, Mar 30 2005

L. Dorst and A. W. M. Smeulders, Discrete straight line segments: parameters, primitives and properties. Vision geometry (Hoboken, NJ, 1989), 45-62, Contemp. Math., 119, Amer. Math. Soc., Providence, RI, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
C. A. Berenstein and D. Lavine, A Geometric Approach to Subpixel Registration Accuracy, CVGIP 40, 1987, 334-360.
C. A. Berenstein and D. Lavine, On the Number of Digital Straight Line Segments, IEEE PAMI, vol.10, no.6, 1988, 880-887
J. Berstel and M. Pocchiola, A geometric proof of the enumeration formula for Sturmian words, Internat. J. Algeb. Comput., 3(3):349-355, 1993.
Michelangelo Bucci, Alessandro De Luca, Amy Glen and Luca Q. Zamboni, A connection between palindromic and factor complexity using return words, arXiv:0802.1332 [math.CO], 2008.
Aldo de Luca and Stefano Varricchio, Finiteness and regularity in semigroups and formal languages, Monographs in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 1999. x+240 pp. ISBN: 3-540-63771-0 MR1696498 (2000g:68001). See p. 25.
L. Dorst, Discrete Straight Line Segments: Parameters, Primitives and Properties, Ph. D. Dissertation, Delft Univ. Technology, 1986. See p. 85.
L. Dorst and A. W. M. Smeulders, Discrete Representation of Straight Lines, IEEE PAMI-6, no.4, 1984, pp. 450-463.
F. Mignosi, On the Number of Factors of Sturmian Words, Theor. Comput. Sci. 82(1): 71-84 (1991)

Cf. A000010, A002088, A011755, A049695, A049703, A103116.

a005598 n = 1 + sum (zipWith (*) [n, n - 1 .. 1] a000010_list)
-- Reinhard Zumkeller, Apr 14 2013

A005598:= func< n | n eq 0 select 1 else 1 +(&+[(n-j+1)*EulerPhi(j): j in [1..n]]) >;
[A005598(n): n in [0..60]]; // G. C. Greubel, Dec 07 2022

f:= m -> add((m-i+1)*phi(i),i=1..m)+1; (Jamet)

Accumulate@Accumulate@EulerPhi@Range[0,100]+1 (* Vladimir Joseph Stephan Orlovsky, Apr 21 2011 *)
Nest[Accumulate[#]&,EulerPhi[Range[0,50]],2]+1 (* Harvey P. Dale, Feb 07 2015 *)

a(n) = 1 + sum(i=1, n, (n-i+1)*eulerphi(i)); \\ Michel Marcus, Aug 04 2016

@CachedFunction
def A005598(n): return 1 + sum( (n-j+1)*euler_phi(j) for j in range(1,n+1) )
[A005598(n) for n in range(61)] # G. C. Greubel, Dec 07 2022

a(n) = 2*A049703(n) for n >= 1.
a(n) = Sum_{i=0..n} A049695(i,n-i). - Clark Kimberling
Asymptotically, a(n) behaves like n^3/Pi^2. - Leo Dorst (leo(AT)science.uva.nl), Apr 02 2007
G.f.: 1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 16 2017
a(n) = 1 + (n+1)*A002088(n) - A011755(n). - G. C. Greubel, Dec 07 2022

Extended by John W. Layman, Mar 30 2005
More terms from Emeric Deutsch, Feb 04 2006
Entry revised by N. J. A. Sloane, Apr 04 2007
Minor English revisions by Jeffrey Shallit, Aug 04 2016

A098198 Decimal expansion of Pi^4/36 = zeta(2)^2.

Original entry on oeis.org

2, 7, 0, 5, 8, 0, 8, 0, 8, 4, 2, 7, 7, 8, 4, 5, 4, 7, 8, 7, 9, 0, 0, 0, 9, 2, 4, 1, 3, 5, 2, 9, 1, 9, 7, 5, 6, 9, 3, 6, 8, 7, 7, 3, 7, 9, 7, 9, 6, 8, 1, 7, 2, 6, 9, 2, 0, 7, 4, 4, 0, 5, 3, 8, 6, 1, 0, 3, 0, 1, 5, 4, 0, 4, 6, 7, 4, 2, 2, 1, 1, 6, 3, 9, 2, 2, 7, 4, 0, 8, 9, 8, 5, 4, 2, 4, 9, 7, 9, 3, 0, 8, 2, 4, 7

Offset: 1

Labos Elemer, Sep 21 2004

			2.70580808427784547879000924135291975693687737979... = 2*A152649 = A013661^2.

Ce Xu and Jianqiang Zhao, Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers, arXiv:2203.04184 [math.NT], 2022.
Index entries for transcendental numbers

Cf. A002088, A024916.

Mathematica
```
RealDigits[N[Pi^4/36, 256]]
```

zeta(2)^2 \\ Charles R Greathouse IV, Aug 08 2013

Decimal expansion of limit of q(n)= A024916(n)/A002088(n) = SummatorySigma / SummatoryTotient.
Equals Sum_{n>=1} A000005(n)/n^2. - R. J. Mathar, Dec 18 2010
Equals 10*Sum_{n>=2} (psi(n)+gamma)/n^3. - Jean-François Alcover, Feb 25 2013
Equals Zeta(4)*10/4 = A013662/0.4 = 1/A227929. - R. J. Mathar, Jul 20 2025
Equals 10 * zeta(3,1) = 10 * Sum_{n >= 1} 1/n Sum_{k >= n+1} 1/k^3 = 10 * Sum_{n >= 1} 1/n^3 * Sum_{k = 1..n-1} 1/k. - Peter Bala, Aug 07 2025

A068773 Alternating sum phi(1) - phi(2) + phi(3) - phi(4) + ... + ((-1)^(n+1))*phi(n).

Original entry on oeis.org

1, 0, 2, 0, 4, 2, 8, 4, 10, 6, 16, 12, 24, 18, 26, 18, 34, 28, 46, 38, 50, 40, 62, 54, 74, 62, 80, 68, 96, 88, 118, 102, 122, 106, 130, 118, 154, 136, 160, 144, 184, 172, 214, 194, 218, 196, 242, 226, 268, 248, 280, 256, 308, 290, 330, 306, 342, 314, 372, 356

Offset: 1

Klaus Brockhaus, Feb 28 2002

			a(3) = phi(1) - phi(2) + phi(3) = 1 - 1 + 2 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000010, A067929.

Cf. A002088, A049690.

with(numtheory): seq(add((-1)^(k+1)*phi(k),k=1..n), n=1..80); # Ridouane Oudra, Mar 22 2024

Accumulate[Array[(-1)^(# + 1) * EulerPhi[#] &, 100]] (* Amiram Eldar, Oct 14 2022 *)

a068773(m)=local(s,n); s=0; for(n=1,m, if(n%2==0,s=s-eulerphi(n),s=s+eulerphi(n)); print1(s,","))

# uses code from A002088 and A049690
def A068773(n): return A002088(n)-(A049690(n>>1)<<1) # Chai Wah Wu, Aug 04 2024

a(n) = Sum_{k=1..n} (-1)^(k+1)*phi(k).
a(n) = n^2/Pi^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022
a(n) = 3*A002088(n) - 2*A049690(n). - Ridouane Oudra, Mar 22 2024
a(n) = A002088(n) - 2*A049690(floor(n/2)). - Chai Wah Wu, Aug 04 2024

A049690 a(n) = Sum_{k=1..n} phi(2*k), where phi = Euler totient function, cf. A000010.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 17, 23, 31, 37, 45, 55, 63, 75, 87, 95, 111, 127, 139, 157, 173, 185, 205, 227, 243, 263, 287, 305, 329, 357, 373, 403, 435, 455, 487, 511, 535, 571, 607, 631, 663, 703, 727, 769, 809, 833, 877, 923, 955, 997, 1037, 1069, 1117, 1169, 1205

Offset: 0

Clark Kimberling

nonn

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

a(n)=b(2n), where b=A049689. Bisections: A099958, A190815.

Cf. A062570.

A049690 := proc(n) return add(numtheory[phi](2*k), k=1..n): end: seq(A049690(n),n=0..54); # Nathaniel Johnston, May 24 2011

A049690[0]:=0; A049690[n_]:=A049690[n-1]+EulerPhi[2n]; Array[A049690,200,0] (* Enrique Pérez Herrero, Feb 25 2012 *)

a(n)=sum(k=1,n,eulerphi(2*k)) \\ Charles R Greathouse IV, Feb 19 2013

from sympy import totient
def A049690(n): return sum(totient(n) for n in range(1,n+1,2)) + (sum(totient(n) for n in range(2,n+1,2))<<1) # Chai Wah Wu, Aug 04 2024

# faster program using program from A002088 and recursive formula
def A049690(n): return A002088(n) + A049690(n>>1) if n else 0 # Chai Wah Wu, Aug 04 2024

a(n) ~ 4*n^2/Pi^2. - Vaclav Kotesovec, Aug 20 2021

a(n) = A002088(n) + a(floor(n/2)). - Chai Wah Wu, Aug 04 2024

More terms from Vladeta Jovovic, May 18 2001

cp's OEIS Frontend

A038567 Denominators in canonical bijection from positive integers to positive rationals <= 1.

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A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

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A020652 Numerators in canonical bijection from positive integers to positive rationals.

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A104141 Decimal expansion of 3/Pi^2.

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A015614 a(n) = -1 + Sum_{i=1..n} phi(i).

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