A065484 -id:A065484 - cp's OEIS frontend

A002618 a(n) = n*phi(n).

1, 2, 6, 8, 20, 12, 42, 32, 54, 40, 110, 48, 156, 84, 120, 128, 272, 108, 342, 160, 252, 220, 506, 192, 500, 312, 486, 336, 812, 240, 930, 512, 660, 544, 840, 432, 1332, 684, 936, 640, 1640, 504, 1806, 880, 1080, 1012, 2162, 768, 2058, 1000

Offset: 1

N. J. A. Sloane

Also Euler phi function of n^2.
For n >= 3, a(n) is also the size of the automorphism group of the dihedral group of order 2n. This automorphism group is isomorphic to the group of transformations x -> ax + b, where a, b and x are integers modulo n and a is coprime to n. Its order is n*phi(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 18 2001
Order of metacyclic group of polynomial of degree n. - Artur Jasinski, Jan 22 2008
It appears that this sequence gives the number of permutations of 1, 2, 3, ..., n that are arithmetic progressions modulo n. - John W. Layman, Aug 27 2008
The conjecture by Layman is correct. Obviously any such permutation must have an increment that is prime to n, and almost as obvious that any such increment will work, with any starting value; hence phi(n) * n total. - Franklin T. Adams-Watters, Jun 09 2009
Consider the numbers from 1 to n^2 written line by line as an n X n square: a(n) = number of numbers that are coprime to all their horizontal and vertical immediate neighbors. - Reinhard Zumkeller, Apr 12 2011
n -> a(n) is injective: a(m) = a(n) implies m = n. - Franz Vrabec, Dec 12 2012 (See Mathematics Stack Exchange link for a proof.)
a(p) = p*(p-1) a pronic number, see A036689 and A002378. - Fred Daniel Kline, Mar 30 2015
Conjecture: All the rational numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015
From Jianing Song, Aug 25 2023: (Start)
a(n) is the order of the holomorph (see the Wikipedia link) of the cyclic group of order n. Note that Hol(C_n) and Aut(D_{2n}) are isomorphic unless n = 2, where D_{2n} is the dihedral group of order 2*n. See the Wordpress link.
The odd-indexed terms form a subsequence of A341298: the holomorph of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link. (End)

			a(4) = 8 since phi(4) = 2 and 4 * 2 = 8.
a(5) = 20 since phi(5) = 4 and 5 * 4 = 20.

Peter Giblin, Primes and Programming: An Introduction to Number Theory with Computing. Cambridge: Cambridge University Press (1993) p. 116, Exercise 1.10.
J. L. Lagrange, Oeuvres, Vol. III Paris 1869.
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).

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Daniel Fischer, answer to Injectivity of the function n times the Euler Totient of n, Mathematics Stack Exchange, October 2013.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Dmitry Krachun and Zhi-Wei Sun, Each positive rational number has the form phi(m^2)/phi(n^2), arXiv:2001.03736 [math.HO], 2020. See also The American Mathematical Monthly (2020) Vol. 127, Issue 9, 847-849.
F. Luca and A. O. Munagi, The number of permutations which form arithmetic progressions modulo m, Annals of the Alexandru Ioan Cuza University, 2014, DOI: 10.2478/aicu-2014-0053. [Broken link]
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Indag. Math. Proc. Ser. A, 60. (1957) 608-619.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]
Wikipedia, Holomorph.
Wordpress, Automorphisms of dihedral groups.

First column of A047916.
Cf. A002619, A011755 (partial sums), A047918, A000010, A053650, A053191, A053192, A036689, A058161, A009262, A082473 (same terms, sorted into ascending order), A256545, A327172 (a left inverse), A327173, A065484.
Subsequence of A323333.

a002618 n = a000010 n * n  -- Reinhard Zumkeller, Dec 21 2012

[n*EulerPhi(n): n in [1..150]]; // Vincenzo Librandi, Apr 04 2011

with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); # Zerinvary Lajos, Oct 07 2007

Table[n EulerPhi[n], {n, 100}] (* Artur Jasinski, Jan 22 2008 *)

numlib::phi(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

a(n)=n*eulerphi(n) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import totient as phi
def a(n): return n*phi(n)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 16 2022

[euler_phi(n^2) for n in range(1,51)] # Zerinvary Lajos, Jun 06 2009

Multiplicative with a(p^e) = (p-1)*p^(2e-1). - David W. Wilson, Aug 01 2001
Dirichlet g.f.: zeta(s-2)/zeta(s-1). - R. J. Mathar, Feb 09 2011
a(n) = A173557(n) * A102631(n). - R. J. Mathar, Mar 30 2011
From Wolfdieter Lang, May 12 2011: (Start)
a(n)/2 = A023896(n), n >= 2.
a(n)/2 = (1/n) * Sum_{k=1..n-1, gcd(k,n)=1} k, n >= 2 (see A023896 and A076512/A109395). (End)
a(n) = lcm(phi(n^2),n). - Enrique Pérez Herrero, May 11 2012
a(n) = phi(n^2). - Wesley Ivan Hurt, Jun 16 2013
a(n) = A009195(n) * A009262(n). - Michel Marcus, Oct 24 2013
G.f.: -x + 2*Sum_{k>=1} mu(k)*k*x^k/(1 - x^k)^3. - Ilya Gutkovskiy, Jan 03 2017
a(n) = A082473(A327173(n)), A327172(a(n)) = n. -- Antti Karttunen, Sep 29 2019
Sum_{n>=1} 1/a(n) = 2.203856... (A065484). - Amiram Eldar, Sep 30 2019
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ c*sqrt(x) for c = Product_{p prime} (1 + 1/(p*(p - 1 + sqrt(p^2 - p)))) = 1.3651304521525857... - Charles R Greathouse IV, Mar 16 2022
a(n) = Sum_{d divides n} A001157(d)*A046692(n/d); that is, the Dirichlet convolution of sigma_2(n) and the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024

Better description from Labos Elemer, Feb 18 2000

A023896 Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention.

Original entry on oeis.org

1, 1, 3, 4, 10, 6, 21, 16, 27, 20, 55, 24, 78, 42, 60, 64, 136, 54, 171, 80, 126, 110, 253, 96, 250, 156, 243, 168, 406, 120, 465, 256, 330, 272, 420, 216, 666, 342, 468, 320, 820, 252, 903, 440, 540, 506, 1081, 384, 1029, 500, 816, 624, 1378, 486, 1100, 672

Offset: 1

Olivier Gérard

Sum of totatives of n, i.e., sum of integers up to n and coprime to n.
a(1) = 1, since 1 is coprime to any positive integer.
Row sums of A038566. - Wolfdieter Lang, May 03 2015
Islam & Manzoor prove that a(n) is an injection for n > 1, see links. In other words, if a(m) = a(n), and min(m, n) > 1, then m = n. - Muhammed Hedayet, May 19 2024

			G.f. = x + x^2 + 3*x^3 + 4*x^4 + 10*x^5 + 6*x^6 + 21*x^7 + 16*x^8 + 27*x^9 + ...
a(12) = 1 + 5 + 7 + 11 = 24.
n = 40: The smallest positive reduced residue system modulo 40 is {1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39}. The sum is a(40) = 320. Average is 20.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n).
David M. Burton, Elementary Number Theory, p. 171.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2001, p. 163.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 111.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
Geoffrey B. Campbell, Dirichlet summations and products over primes, Int. J. Math. Math. Sci. 16 92) (1993) 359. eq. (3.1)
Muhammed H. Islam and Shahriar Manzoor, φ1 and phitorial are injections, for any positive integer N, where N > 1.
Constantin M. Petridi, The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots, arXiv:1612.07632 [math.NT], 2016.
David Zmiaikou, Origamis and permutation groups, Thesis, 2011. See p. 65.

Cf. A000010, A000203, A002180, A045545, A001783, A024816, A066760, A054521, A067392, A038566.
Row sums of A127368, A144734, A144824.
Cf. A000217, A002618, A008683, A009195, A023022, A065484, A076512, A175505, A175506.

a023896 = sum . a038566_row  -- Reinhard Zumkeller, Mar 04 2012

[1] cat [n*EulerPhi(n)/2: n in [2..70]]; // Vincenzo Librandi, May 16 2015

A023896 := proc(n)
    if n = 1 then
        1;
    else
        n*numtheory[phi](n)/2 ;
    end if;
end proc: # R. J. Mathar, Sep 26 2013

a[ n_ ] = n/2*EulerPhi[ n ]; a[ 1 ] = 1; Table[a[n], {n, 56}]
a[ n_] := If[ n < 2, Boole[n == 1], Sum[ k Boole[1 == GCD[n, k]], { k, n}]]; (* Michael Somos, Jul 08 2014 *)

{a(n) = if(n<2, n>0, n*eulerphi(n)/2)};

A023896(n)=n*eulerphi(n)\/2 \\ about 10% faster. - M. F. Hasler, Feb 01 2021

from sympy import totient
def A023896(n): return 1 if n == 1 else n*totient(n)//2 # Chai Wah Wu, Apr 08 2022

def A023896(n): return 1 if n == 1 else n*euler_phi(n)//2
print([A023896(n) for n in range(1, 57)])  # Peter Luschny, Dec 03 2023

a(n) = n*A023022(n) for n > 2.
a(n) = phi(n^2)/2 = n*phi(n)/2 = A002618(n)/2 if n > 1, a(1)=1. See the Apostol reference for this exercise.
a(n) = Sum_{1 <= k < n, gcd(k, n) = 1} k.
If n = p is a prime, a(p) = T(p-1) where T(k) is the k-th triangular number (A000217). - Robert G. Wilson v, Jul 31 2004
Equals A054521 * [1,2,3,...]. - Gary W. Adamson, May 20 2007
a(n) = A053818(n) * A175506(n) / A175505(n). - Jaroslav Krizek, Aug 01 2010
If m,n > 1 and gcd(m,n) = 1 then a(m*n) = 2*a(m)*a(n). - Thomas Ordowski, Nov 09 2014
G.f.: Sum_{n>=1} mu(n)*n*x^n/(1-x^n)^3, where mu(n) = A008683(n). - Mamuka Jibladze, Apr 24 2015
G.f. A(x) satisfies A(x) = x/(1 - x)^3 - Sum_{k>=2} k * A(x^k). - Ilya Gutkovskiy, Sep 06 2019
For n > 1: a(n) = (n*A076512(n)/2)*A009195(n). - Jamie Morken, Dec 16 2019
Sum_{n>=1} 1/a(n) = 2 * A065484 - 1 = 3.407713... . - Amiram Eldar, Oct 09 2023

Typos in programs corrected by Zak Seidov, Aug 03 2010

Name and example edited by Wolfdieter Lang, May 03 2015

A272030 Decimal expansion of C = log(2*Pi) + B_3 (where B_3 is A083343), one of Euler totient constants.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 25 2016

			3.17045934214256636532648824888226302856125443631798948742143398...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 117.

Cf. A001620, A005596, A065484, A065485, A080130, A082695, A082695, A083343.

digits = 98; B3 = EulerGamma - NSum[PrimeZetaP'[n], {n, 2, Infinity}, WorkingPrecision -> 2 digits, NSumTerms -> 200]; RealDigits[Log[2 Pi] + B3, 10, digits][[1]]

C = log(2*Pi) + EulerGamma - Sum_{n >= 2} P'(n), where P'(n) is the prime zeta P function derivative.

A065483 Decimal expansion of totient constant Product_{p prime} (1 + 1/(p^2*(p-1))).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

The sum of the reciprocals of the cubefull numbers (A036966). - Amiram Eldar, Jun 23 2020

			1.339784153574347246599152586514886052775...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Eric Weisstein's World of Mathematics, Totient Summatory Function

Cf. A036966, A065484, A078074.

$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0}, LinearRecurrence[{2, -1, -1, 1}, {3, 4, 5, 3}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p^2*(p-1))) \\ Vaclav Kotesovec, Sep 19 2020

Equals (6/Pi^2) * A065484. - Amiram Eldar, Jun 23 2020

A008332 Sum of divisors of p-1, p prime.

Original entry on oeis.org

1, 3, 7, 12, 18, 28, 31, 39, 36, 56, 72, 91, 90, 96, 72, 98, 90, 168, 144, 144, 195, 168, 126, 180, 252, 217, 216, 162, 280, 248, 312, 252, 270, 288, 266, 372, 392, 363, 252, 308, 270, 546, 360, 508, 399, 468, 576, 456, 342, 560, 450, 432, 744, 468, 511, 396, 476, 720, 672

Offset: 1

N. J. A. Sloane

For all n (except for n = 2) gcd(A008332(n), prime(n)) = 1. - Lechoslaw Ratajczak, Aug 22 2018

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 87.

Michel Marcus, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
William A. Webb, An asymptotic estimate for a class of divisor sums, Duke Math. J., Vol. 38, No. 3 (1971), pp. 575-582.

Cf. A000203, A006093, A008332, A065484.

[DivisorSigma(1, NthPrime(n)-1): n in [1..60]]; // Vincenzo Librandi, Aug 20 2018

for i from 1 to 500 do if isprime(i) then print(sigma(i-1)); fi; od;

Table[DivisorSigma[1, Prime[n] - 1], {n, 80}] (* Vincenzo Librandi, Aug 20 2018 *)

a(n) = sigma(prime(n)-1); \\ Michel Marcus, Aug 19 2018

a(n) = A000203(A006093(n)). - Michel Marcus, Aug 19 2018

Sum_{k; prime(k)<=x} a(k) ~ c * x^2/(2*log(x)), where c = A065484 (Webb, 1971). - Amiram Eldar, Mar 04 2021 [corrected Jul 20 2025]

Offset corrected by Michel Marcus, Aug 20 2018

A109695 Decimal expansion of Sum_{n>=1} 1/phi(n)^2.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Aug 07 2005

The logarithm of the value can be expanded in a series Sum_{j>=2} c(j)*P(j) = P(2) + 2*P(3) + (7/2)*P(4) + ... where P(.) is the prime zeta function. The partial sums of the series are a slowly oscillating function of the upper limit of j, from which the bracketing interval [3.390642005572503655..., 3.390642005572504756...] for the constant can be computed. - R. J. Mathar, Feb 03 2009

Sum_{n>=1} 1/phi(n)^k is convergent iff k > 1 (reference Monier). - Bernard Schott, Dec 13 2020

			3.39064200557250391614259566300263079374053738121447169118...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21, pp. 281 and 294.

Cf. A000010, A065484, A127473, A335818.

$MaxExtraPrecision = 1000; f[p_] := (1 + p^2/((p - 1)^2*(p^2 - 1))); Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 25 2020 *)

my(N=1000000000); prodeuler(p=2,N,1.+p^2/((p-1)^2*(p^2-1)))*(1+1/(N*log(N)))

prodeulerrat(1 + p^2/((p-1)^2*(p^2-1))) \\ Amiram Eldar, Mar 15 2021

Equals Product_p Sum_{k>=0} 1/phi(p^k)^2 = Product_p (1 + p^2/((p-1)^2*(p^2-1))).

Equals Sum{n>=1} 1/A127473(n). - Amiram Eldar, Mar 15 2021

Four more digits from R. J. Mathar, Feb 03 2009, 25 more Dec 18 2010

More digits from Vaclav Kotesovec, Jun 25 2020

A335818 Decimal expansion of Sum_{k>=1} 1/phi(k)^3, where phi is the Euler totient function.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 25 2020

Sum_{k>=1} 1/phi(k)^m is convergent iff m > 1 (reference Monier). - Bernard Schott, Jan 14 2021

			2.476194748165025794326855444125145160045456856355284384345707879150949...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21, pp. 281 and 294.

Eric Weisstein's World of Mathematics, Totient Function.
Eric Weisstein's World of Mathematics, Totient Summatory Function.
Wikipedia, Euler's totient function.

Cf. A000010, A065484, A109695, A335319.

$MaxExtraPrecision = 1000; f[p_] := (1 + 1/((1 - 1/p)^s * (p^s - 1))) /. s -> 3; Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {m, 100, 1000, 100}]

prodeulerrat(1 + 1/((1 - 1/p)^3 * (p^3 - 1))) \\ Amiram Eldar, Mar 15 2021

Equals Product_{primes p} (1 + 1/((1 - 1/p)^3 * (p^3 - 1))).

A335319 Decimal expansion of Sum_{n>=2} (-1)^n/(n*phi(n)), where phi(n) is the Euler totient function A000010.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, May 31 2020

The formula section of A000010 provides the following conjecture: Sum_{i>=2} (-1)^i/(i*phi(i)) exists and is approximately 0.558. - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004

A more accurate value of the conjectured limit is provided.

			0.5592286807124280424254343367039820674865653612424...

Cf. A000010, A002618, A065484.

1 - prodeulerrat(1 + p/((p-1)^2*(p+1)))/5 \\ Amiram Eldar, Nov 11 2020

Equals 1 - (1/5) * A065484. - Amiram Eldar, Nov 11 2020

More terms from Amiram Eldar, Nov 11 2020

A335762 Decimal expansion of Product_{p prime} (1 + p/((p-1)*(p+1)^2)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 21 2020

The asymptotic mean of A367987. - Amiram Eldar, Dec 23 2023

			1.450032145362120831608395887189223422325062117447167...

Cf. A000082, A001615, A013661, A065465, A065484, A367987.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-1, 1, 2, 0, -1}, {0, 2, -3, 6, -5}, m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 + p/((p-1)*(p+1)^2)) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{k>=1} 1/A000082(k) = Sum_{k>=1} 1/(k * A001615(k)).

Equals A013661 * A065465. - Amiram Eldar, Dec 23 2023

More digits from Vaclav Kotesovec, Sep 19 2020

cp's OEIS Frontend

A118262 Duplicate of A065484.

Views

Author

Keywords

A002618 a(n) = n*phi(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

MuPAD

PARI

Python

Sage

Formula

Extensions

A023896 Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

SageMath

Formula

Extensions

A272030 Decimal expansion of C = log(2*Pi) + B_3 (where B_3 is A083343), one of Euler totient constants.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A065483 Decimal expansion of totient constant Product_{p prime} (1 + 1/(p^2*(p-1))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A008332 Sum of divisors of p-1, p prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI