A306633 -id:A306633 - cp's OEIS frontend

A000010 Euler totient function phi(n): count numbers <= n and prime to n.

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44

Offset: 1

N. J. A. Sloane

Number of elements in a reduced residue system modulo n.
Degree of the n-th cyclotomic polynomial (cf. A013595). - Benoit Cloitre, Oct 12 2002
Number of distinct generators of a cyclic group of order n. Number of primitive n-th roots of unity. (A primitive n-th root x is such that x^k is not equal to 1 for k = 1, 2, ..., n - 1, but x^n = 1.) - Lekraj Beedassy, Mar 31 2005
Also number of complex Dirichlet characters modulo n; Sum_{k=1..n} a(k) is asymptotic to (3/Pi^2)*n^2. - Steven Finch, Feb 16 2006
a(n) is the highest degree of irreducible polynomial dividing 1 + x + x^2 + ... + x^(n-1) = (x^n - 1)/(x - 1). - Alexander Adamchuk, Sep 02 2006, corrected Sep 27 2006
a(p) = p - 1 for prime p. a(n) is even for n > 2. For n > 2, a(n)/2 = A023022(n) = number of partitions of n into 2 ordered relatively prime parts. - Alexander Adamchuk, Jan 25 2007
Number of automorphisms of the cyclic group of order n. - Benoit Jubin, Aug 09 2008
a(n+2) equals the number of palindromic Sturmian words of length n which are "bispecial", prefix or suffix of two Sturmian words of length n + 1. - Fred Lunnon, Sep 05 2010
Suppose that a and n are coprime positive integers, then by Euler's totient theorem, any factor of n divides a^phi(n) - 1. - Lei Zhou, Feb 28 2012
If m has k prime factors, (p_1, p_2, ..., p_k), then phi(m*n) = (Product_{i=1..k} phi (p_i*n))/phi(n)^(k-1). For example, phi(42*n) = phi(2*n)*phi(3*n)*phi(7*n)/phi(n)^2. - Gary Detlefs, Apr 21 2012
Sum_{n>=1} a(n)/n! = 1.954085357876006213144... This sum is referenced in Plouffe's inverter. - Alexander R. Povolotsky, Feb 02 2013 (see A336334. - Hugo Pfoertner, Jul 22 2020)
The order of the multiplicative group of units modulo n. - Michael Somos, Aug 27 2013
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 30 2016
From Eric Desbiaux, Jan 01 2017: (Start)
a(n) equals the Ramanujan sum c_n(n) (last term on n-th row of triangle A054533).
a(n) equals the Jordan function J_1(n) (cf. A007434, A059376, A059377, which are the Jordan functions J_2, J_3, J_4, respectively). (End)
For n > 1, a(n) appears to be equal to the number of semi-meander solutions for n with top arches containing exactly 2 mountain ranges and exactly 2 arches of length 1. - Roger Ford, Oct 11 2017
a(n) is the minimum dimension of a lattice able to generate, via cut-and-project, the quasilattice whose diffraction pattern features n-fold rotational symmetry. The case n=15 is the first n > 1 in which the following simpler definition fails: "a(n) is the minimum dimension of a lattice with n-fold rotational symmetry". - Felix Flicker, Nov 08 2017
Number of cyclic Latin squares of order n with the first row in ascending order. - Eduard I. Vatutin, Nov 01 2020
a(n) is the number of rational numbers p/q >= 0 (in lowest terms) such that p + q = n. - Rémy Sigrist, Jan 17 2021
From Richard L. Ollerton, May 08 2021: (Start)
Formulas for the numerous OEIS entries involving Dirichlet convolution of a(n) and some sequence h(n) can be derived using the following (n >= 1):
Sum_{d|n} phi(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k)) [see P. H. van der Kamp link] = Sum_{d|n} h(d)*phi(n/d) = Sum_{k=1..n} h(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). Similarly,
Sum_{d|n} phi(d)*h(d) = Sum_{k=1..n} h(n/gcd(n,k)) = Sum_{k=1..n} h(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).
More generally,
Sum_{d|n} h(d) = Sum_{k=1..n} h(gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))/phi(n/gcd(n,k)).
In particular, for sequences involving the Möbius transform:
Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where mu = A008683.
Use of gcd(n,k)*lcm(n,k) = n*k and phi(gcd(n,k))*phi(lcm(n,k)) = phi(n)*phi(k) provide further variations. (End)
From Richard L. Ollerton, Nov 07 2021: (Start)
Formulas for products corresponding to the sums above may found using the substitution h(n) = log(f(n)) where f(n) > 0 (for example, cf. formulas for the sum A018804 and product A067911 of gcd(n,k)):
Product_{d|n} f(n/d)^phi(d) = Product_{k=1..n} f(gcd(n,k)) = Product_{d|n} f(d)^phi(n/d) = Product_{k=1..n} f(n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
Product_{d|n} f(d)^phi(d) = Product_{k=1..n} f(n/gcd(n,k)) = Product_{k=1..n} f(gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
Product_{d|n} f(d) = Product_{k=1..n} f(gcd(n,k))^(1/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(1/phi(n/gcd(n,k))),
Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))), where mu = A008683. (End)
a(n+1) is the number of binary words with exactly n distinct subsequences (when n > 0). - Radoslaw Zak, Nov 29 2021

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 6*x^7 + 4*x^8 + 6*x^9 + 4*x^10 + ...
a(8) = 4 with {1, 3, 5, 7} units modulo 8. a(10) = 4 with {1, 3, 7, 9} units modulo 10. - _Michael Somos_, Aug 27 2013
From _Eduard I. Vatutin_, Nov 01 2020: (Start)
The a(5)=4 cyclic Latin squares with the first row in ascending order are:
  0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
  1 2 3 4 0   2 3 4 0 1   3 4 0 1 2   4 0 1 2 3
  2 3 4 0 1   4 0 1 2 3   1 2 3 4 0   3 4 0 1 2
  3 4 0 1 2   1 2 3 4 0   4 0 1 2 3   2 3 4 0 1
  4 0 1 2 3   3 4 0 1 2   2 3 4 0 1   1 2 3 4 0
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
M. Baake and U. Grimm, Aperiodic Order Vol. 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications 149, Cambridge University Press, 2013: see Tables 3.1 and 3.2.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 409.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 154-156.
C. W. Curtis, Pioneers of Representation Theory ..., Amer. Math. Soc., 1999; see p. 3.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, Paris, 2004, Problème 529, pp. 71-257.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, Chapter V.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
Carl Friedrich Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965; see p. 21.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, p. 137.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B36.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 60, 62, 63, 288, 323, 328, 330.
Peter Hilton and Jean Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, pages 261-264, the Coach theorem.
Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21 pp. 281-294.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 2 vols., 1976, Vol. II, problem 71, p. 126.
Paulo Ribenboim, The New Book of Prime Number Records.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 28-33.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 162-167.

Daniel Forgues, Table of n, phi(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
Dario A. Alpern, Factorization using the Elliptic Curve Method (along with sigma_0, sigma_1 and phi functions)
Joerg Arndt, Matters Computational (The Fxtbook), section 39.7, pp. 776-778.
F. Bayart, Indicateur d'Euler (in French).
Alexander Bogomolny, Euler Function and Theorem.
Chris K. Caldwell, The Prime Glossary, Euler's phi function
Robert D. Carmichael, A table of the values of m corresponding to given values of phi(m), Amer. J. Math., 30 (1908), 394-400. [Annotated scanned copy]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Kevin Ford, The number of solutions of phi(x)=m, arXiv:math/9907204 [math.NT], 1999.
Kevin Ford, Florian Luca and Pieter Moree, Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv:1108.3805 [math.NT], 2011.
H. Fripertinger, The Euler phi function.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
E. Pérez Herrero, Totient Carnival partitions, Psychedelic Geometry Blogspot.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, arXiv:1201.3139 [math.NT]
M. Lal and P. Gillard, Table of Euler's phi function, n < 10^5, Math. Comp., 23 (1969), 682-683.
Derrick N. Lehmer, Review of Dickson's History of the Theory of Numbers, Bull. Amer. Math. Soc., 26 (1919), 125-132.
Peter Luschny, Sequences related to Euler's totient function.
R. J. Mathar, Graphical representation among sequences closely related to this one (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
Mathematics Stack Exchange, Is the Euler phi function bounded below? (2013).
Mathforum, Proving phi(m) Is Even.
Keith Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n).
Graeme McRae, Euler's Totient Function.
François Nicolas, A simple, polynomial-time algorithm for the matrix torsion problem, arXiv:0806.2068 [cs.DM], 2009.
Matthew Parker, The first 5 million terms (7-Zip compressed file).
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Primefan, Euler's Totient Function Values For n=1 to 500, with Divisor Lists.
Marko Riedel, Combinatorics and number theory page.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.
K. Schneider, Euler phi-function, PlanetMath.org.
Wacław F. Sierpiński, Euler's Totient Function And The Theorem Of Euler.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 14.
Ulrich Sondermann, Euler's Totient Function.
William A. Stein, Phi is a Multiplicative Function
Pinthira Tangsupphathawat, Takao Komatsu and Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
G. Villemin, Totient d'Euler.
K. W. Wegner, Values of phi(x) = n for n from 2 through 1978, mimeographed manuscript, no date. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
Eric Weisstein's World of Mathematics, Moebius Transform.
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.
Wikipedia, Multiplicative group of integers modulo n.
Wikipedia, Ramanujan's sum
Wolfram Research, First 50 values of phi(n).
Gang Xiao, Numerical Calculator, To display phi(n) operate on "eulerphi(n)".
Index entries for "core" sequences
Index to divisibility sequences

Cf. A002088 (partial sums), A008683, A003434 (steps to reach 1), A007755, A049108, A002202 (values), A011755 (Sum k*phi(k)).
Cf. also A005277 (nontotient numbers). For inverse see A002181, A006511, A058277.
Jordan function J_k(n) is a generalization - see A059379 and A059380 (triangle of values of J_k(n)), this sequence (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A054521, A023022, A054525.
Row sums of triangles A134540, A127448, A143239, A143353 and A143276.
Equals right and left borders of triangle A159937. - Gary W. Adamson, Apr 26 2009
Values for prime powers p^e: A006093 (e=1), A036689 (e=2), A135177 (e=3), A138403 (e=4), A138407 (e=5), A138412 (e=6).
Values for perfect powers n^e: A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), A306411 (e=6), A239442 (e=7), A306412 (e=8), A239443 (e=9).
Cf. A003558, A135303.
Cf. A152455, A080737.
Cf. A076479.
Cf. A023900 (Dirichlet inverse of phi), A306633 (Dgf at s=3).

Axiom
```
[eulerPhi(n) for n in 1..100]
```

a n = length (filter (==1) (map (gcd n) [1..n])) -- Allan C. Wechsler, Dec 29 2014

# Computes the first N terms of the sequence.
function A000010List(N)
    phi = [i for i in 1:N + 1]
    for i in 2:N + 1
        if phi[i] == i
            for j in i:i:N + 1
                phi[j] -= div(phi[j], i)
    end end end
return phi end
println(A000010List(68))  # Peter Luschny, Sep 03 2023

[ EulerPhi(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

with(numtheory): A000010 := phi; [ seq(phi(n), n=1..100) ]; # version 1
with(numtheory): phi := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := n*mul((1-1/t1[i][1]),i=1..nops(t1)); end; # version 2
# Alternative without library function:
A000010List := proc(N) local i, j, phi;
    phi := Array([seq(i, i = 1 .. N+1)]);
    for i from 2 to N + 1 do
        if phi[i] = i then
            for j from i by i to N + 1 do
                phi[j] := phi[j] - iquo(phi[j], i) od
        fi od;
return phi end:
A000010List(68);  # Peter Luschny, Sep 03 2023

Mathematica
```
Array[EulerPhi, 70]
```

makelist(totient(n),n,0,1000); /* Emanuele Munarini, Mar 26 2011 */

{a(n) = if( n==0, 0, eulerphi(n))}; /* Michael Somos, Feb 05 2011 */

from sympy.ntheory import totient
print([totient(i) for i in range(1, 70)])  # Indranil Ghosh, Mar 17 2017

# Note also the implementation in A365339.

def A000010(n): return euler_phi(n) # Jaap Spies, Jan 07 2007

[euler_phi(n) for n in range(1, 70)]  # Zerinvary Lajos, Jun 06 2009

phi(n) = n*Product_{distinct primes p dividing n} (1 - 1/p).
Sum_{d divides n} phi(d) = n.
phi(n) = Sum_{d divides n} mu(d)*n/d, i.e., the Moebius transform of the natural numbers; mu() = Moebius function A008683().
Dirichlet generating function Sum_{n>=1} phi(n)/n^s = zeta(s-1)/zeta(s). Also Sum_{n >= 1} phi(n)*x^n/(1 - x^n) = x/(1 - x)^2.
Multiplicative with a(p^e) = (p - 1)*p^(e-1). - David W. Wilson, Aug 01 2001
Sum_{n>=1} (phi(n)*log(1 - x^n)/n) = -x/(1 - x) for -1 < x < 1 (cf. A002088) - Henry Bottomley, Nov 16 2001
a(n) = binomial(n+1, 2) - Sum_{i=1..n-1} a(i)*floor(n/i) (see A000217 for inverse). - Jon Perry, Mar 02 2004
It is a classical result (certainly known to Landau, 1909) that lim inf n/phi(n) = 1 (taking n to be primes), lim sup n/(phi(n)*log(log(n))) = e^gamma, with gamma = Euler's constant (taking n to be products of consecutive primes starting from 2 and applying Mertens' theorem). See e.g. Ribenboim, pp. 319-320. - Pieter Moree, Sep 10 2004
a(n) = Sum_{i=1..n} |k(n, i)| where k(n, i) is the Kronecker symbol. Also a(n) = n - #{1 <= i <= n : k(n, i) = 0}. - Benoit Cloitre, Aug 06 2004 [Corrected by Jianing Song, Sep 25 2018]
Conjecture: Sum_{i>=2} (-1)^i/(i*phi(i)) exists and is approximately 0.558 (A335319). - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004
From Enrique Pérez Herrero, Sep 07 2010: (Start)
a(n) = Sum_{i=1..n} floor(sigma_k(i*n)/sigma_k(i)*sigma_k(n)), where sigma_2 is A001157.
a(n) = Sum_{i=1..n} floor(tau_k(i*n)/tau_k(i)*tau_k(n)), where tau_3 is A007425.
a(n) = Sum_{i=1..n} floor(rad(i*n)/rad(i)*rad(n)), where rad is A007947. (End)
a(n) = A173557(n)*A003557(n). - R. J. Mathar, Mar 30 2011
a(n) = A096396(n) + A096397(n). - Reinhard Zumkeller, Mar 24 2012
phi(p*n) = phi(n)*(floor(((n + p - 1) mod p)/(p - 1)) + p - 1), for primes p. - Gary Detlefs, Apr 21 2012
For odd n, a(n) = 2*A135303((n-1)/2)*A003558((n-1)/2) or phi(n) = 2*c*k; the Coach theorem of Pedersen et al. Cf. A135303. - Gary W. Adamson, Aug 15 2012
G.f.: Sum_{n>=1} mu(n)*x^n/(1 - x^n)^2, where mu(n) = A008683(n). - Mamuka Jibladze, Apr 05 2015
a(n) = n - cototient(n) = n - A051953(n). - Omar E. Pol, May 14 2016
a(n) = lim_{s->1} n*zeta(s)*(Sum_{d divides n} A008683(d)/(e^(1/d))^(s-1)), for n > 1. - Mats Granvik, Jan 26 2017
Conjecture: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 for n > 1. The sum is over a,b,c such that n*c - a*b = 1. - Benedict W. J. Irwin, Apr 03 2017
a(n) = Sum_{j=1..n} gcd(j, n) cos(2*Pi*j/n) = Sum_{j=1..n} gcd(j, n) exp(2*Pi*i*j/n) where i is the imaginary unit. Notice that the Ramanujan's sum c_n(k) := Sum_{j=1..n, gcd(j, n) = 1} exp(2*Pi*i*j*k/n) gives a(n) = Sum_{k|n} k*c_(n/k)(1) = Sum_{k|n} k*mu(n/k). - Michael Somos, May 13 2018
G.f.: x*d/dx(x*d/dx(log(Product_{k>=1} (1 - x^k)^(-mu(k)/k^2)))), where mu(n) = A008683(n). - Mamuka Jibladze, Sep 20 2018
a(n) = Sum_{d|n} A007431(d). - Steven Foster Clark, May 29 2019
G.f. A(x) satisfies: A(x) = x/(1 - x)^2 - Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019
a(n) >= sqrt(n/2) (Nicolas). - Hugo Pfoertner, Jun 01 2020
a(n) > n/(exp(gamma)*log(log(n)) + 5/(2*log(log(n)))), except for n=223092870 (Rosser, Schoenfeld). - Hugo Pfoertner, Jun 02 2020
From Bernard Schott, Nov 28 2020: (Start)
Sum_{m=1..n} 1/a(m) = A028415(n)/A048049(n) -> oo when n->oo.
Sum_{n >= 1} 1/a(n)^2 = A109695.
Sum_{n >= 1} 1/a(n)^3 = A335818.
Sum_{n >= 1} 1/a(n)^k is convergent iff k > 1.
a(2n) = a(n) iff n is odd, and, a(2n) > a(n) iff n is even. (End) [Actually, a(2n) = 2*a(n) for even n. - Jianing Song, Sep 18 2022]
a(n) = 2*A023896(n)/n, n > 1. - Richard R. Forberg, Feb 03 2021
From Richard L. Ollerton, May 09 2021: (Start)
For n > 1, Sum_{k=1..n} phi^{(-1)}(n/gcd(n,k))*a(gcd(n,k))/a(n/gcd(n,k)) = 0, where phi^{(-1)} = A023900.
For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(gcd(n,k)))*rad(gcd(n,k))/gcd(n,k) = 0.
For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(n/gcd(n,k)))*rad(n/gcd(n,k))*gcd(n,k) = 0.
Sum_{k=1..n} a(gcd(n,k))/a(n/gcd(n,k)) = n. (End)
a(n) = Sum_{d|n, e|n} gcd(d, e)*mobius(n/d)*mobius(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value p^e - p^(e-1) for n = p^e, a prime power). - Peter Bala, Jan 22 2024
Sum_{n >= 1} phi(n)*x^n/(1 + x^n) = x + 3*x^3 + 5*x^5 + 7*x^7 + ... = Sum_{n >= 1} phi(2*n-1)*x^(2*n-1)/(1 - x^(4*n-2)). For the first equality see Pólya and Szegő, problem 71, p. 126. - Peter Bala, Feb 29 2024
Conjecture: a(n) = lim_{k->oo} (n^(k + 1))/A000203(n^k). - Velin Yanev, Dec 04 2024 [A000010(p) = p-1, A000203(p^k) = (p^(k+1)-1)/(p-1), so the conjecture is true if n is prime. - Vaclav Kotesovec, Dec 19 2024]

A065463 Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

The density of A268335. - Vladimir Shevelev, Feb 01 2016

The probability that two numbers are coprime given that one of them is coprime to a randomly chosen third number. - Luke Palmer, Apr 27 2019

			0.7044422009991655927366033503...

Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013), Article 13.6.3.
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
David Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013-2017.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arxiv:0903.2514 [math.NT] (2009) constant Q_1^(1).
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
R. Sitaramachandrarao and D. Suryanarayana, On Sigma_{n<=x} sigma*(n) and Sigma_{n<=x} phi*(n), Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1,arXiv preprint, arXiv:1608.00795 [math.NT], 2016.
Deyu Zhang and Wenguang Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq., Vol. 13 (2010), Article 10.4.7, eq. (4).
Rimer Zurita Generalized Alternating Sums of Multiplicative Arithmetic Functions, J. Int. Seq., Vol. 23 (2020), Article 20.10.4.

Cf. A001615, A007947, A047994, A078082, A268335, A306633.

Cf. A000203, A065473, A065480, A065490, A008683.

$MaxExtraPrecision = 1200; digits = 98; terms = 1200; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, 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*(p+1))) \\ Amiram Eldar, Mar 14 2021

From Amiram Eldar, Mar 05 2019: (Start)
Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} rad(k), where rad(k) = A007947(k) is the squarefree kernel of k (Cohen).
Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} uphi(k), where uphi(k) = A047994(k) is the unitary totient function (Sitaramachandrarao and Suryanarayana).
Equals lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/psi(k), where psi(k) = A001615(k) is the Dedekind psi function (Sita Ramaiah and Suryanarayana).
(End)
Equals A065473*A013661/A065480. - Luke Palmer, Apr 27 2019
Equals Sum_{k>=1} mu(k)/(k*sigma(k)), where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022
Equals 1/A065489. - R. J. Mathar, May 27 2025

A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 8, 1, 10, 9, 1, 1, 12, 1, 10, 11, 14, 1, 12, 1, 16, 1, 12, 1, 42, 1, 1, 15, 20, 13, 14, 1, 22, 17, 14, 1, 54, 1, 16, 15, 26, 1, 20, 1, 28, 21, 18, 1, 30, 17, 16, 23, 32, 1, 60, 1, 34, 17, 1, 19, 78, 1, 22, 27, 74, 1, 18, 1, 40, 29, 24, 19, 90, 1, 22, 1, 44

Offset: 1

N. J. A. Sloane

			Unitary divisors of 12 are 1, 3, 4, 12. a(12) = 1 + 3 + 4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation 83.288 (2014): 1903-1913; alternative link.

Cf. A034444, A034448, A077610, A306633.
Cf. A063936 (squares > 1).
Cf. A063919 (essentially the same sequence).

a034460 = sum . init . a077610_row  -- Reinhard Zumkeller, Aug 15 2012

A034460 := proc(n)
    A034448(n)-n ;
end proc:
seq(A034460(n),n=1..40) ; # R. J. Mathar, Nov 10 2014

usigma[n_] := Sum[ If[GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; a[n_] := usigma[n] - n; Table[ a[n], {n, 1, 82}] (* Jean-François Alcover, May 15 2012 *)
a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 03 2022 *)

a(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n \\ Charles R Greathouse IV, Aug 01 2016

a(n) = Sum_{k = 1..A034444(n)-1} A077610(n,k). - Reinhard Zumkeller, Aug 15 2012

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/2 = 0.1842163888... . - Amiram Eldar, Feb 22 2024

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

A046125 Number of negative fundamental discriminants having class number n.

Original entry on oeis.org

9, 18, 16, 54, 25, 51, 31, 131, 34, 87, 41, 206, 37, 95, 68, 322, 45, 150, 47, 350, 85, 139, 68, 511, 95, 190, 93, 457, 83, 255, 73, 708, 101, 219, 103, 668, 85, 237, 115, 912, 109, 339, 106, 691, 154, 268, 107, 1365, 132, 345, 159, 770, 114, 427, 163, 1205, 179, 291

Offset: 1

Eric W. Weisstein

			a(1) = 9 because the discriminants {-3,-4,-7,-8,-11,-19,-43,-67,-163} are the only ones with class number 1.

Andy Huchala, Table of n, a(n) for n = 1..100 (from Watkins' paper)
Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.
K. Soundararajan, The number of imaginary quadratic fields with a given class number, Hardy-Ramanujan Journal, Vol. 30 (2007), pp. 13-18.
Mark Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation 73 (2004), pp. 907-938.
Eric Weisstein's World of Mathematics, Class Number.
Index entries for sequences related to quadratic fields.

Cf. A014602, A014603, A006203, A013658, A046002-A046016, A048925, A056987, A081319, A038552, A191408, A191410.

Cf. A038552, A306633.

FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *);
k = 1; t = Table[0, {125}]; While[k < 2000001, If[ FundamentalDiscriminantQ@ -k, a = NumberFieldClassNumber@ Sqrt@ -k; If[a < 126, t[[a]]++]]; k++]; t (* Robert G. Wilson v Jun 01 2011 *)

lista(nn=10^7) = {my(NMAX=100, v = vector(NMAX), c); for (k=1, nn, if (isfundamental(-k), if ((c = qfbclassno(-k)) <= NMAX, v[c] ++););); v;} \\ Michel Marcus, Feb 17 2022

From Amiram Eldar, Apr 15 2025: (Start)
Formulas from Soundararajan (2007):
Sum_{k=1..n} a(k) = (3*zeta(2)/zeta(3)) * n^2 + O(n^2 * log(n)^(-1/2+eps)).
a(n) << n^2 * log(log(n))^4 / log(n). (End)

Edited by Robert G. Wilson v, May 13 2003

Corrected and extended by Dean Hickerson, May 20 2003. The values were obtained by transcribing and combining data from Tables 1-3 of Buell's paper, which has information for all values of n up to 125.

A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

Original entry on oeis.org

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 10, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 18, 48, 54, 48, 31, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 40, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 34, 84, 144, 68, 72, 96, 144, 72, 51, 74, 114, 64, 80, 96, 168, 80, 60, 43, 126

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

This is not multiplicative: a(4) = 4, a(9) = 7, but a(36) = 31, not 28. However, the function acts multiplicatively on certain subsequences of natural numbers, like for example when restricted to A048107, where this sequence coincides with A326043.

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, unitary divisors are 1, 4, 9 and 36, so A034448(36) = 1+4+9+36 = 50, while the squarefree divisors are 1, 2, 3 and 6, so A048250(36) = 1+2+3+6 = 12, thus a(36) = (50+12)/2 = 31.
For n = 495, its divisors are 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495. Of these, unitary are 1, 5, 9, 11, 45, 55, 99, 495, whose sum is A034448(495) = 720, while the squarefree divisors are 1, 3, 5, 11, 15, 33, 55, 165, and their sum is A048250(495) = 288. Thus a(495) = (720+288)/2 = 504. Also for 495, whose prime factorization is 3^2 * 5^1 * 11^1 this can be computed faster as the average of ((3^2)+1)*(5+1)*(11+1) and (3+1)*(5+1)*(11+1), thus (1/2)*(3+(3^2)+2)*(5+1)*(11+1) = 504.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A034448, A048107, A048250, A052548, A289521, A306633, A325974, A325975, A325977, A325978, A325981, A326043.

Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #) + Times @@ (1 + #[[;; , 1]]) &@ FactorInteger[#]] &, 90] (* Michael De Vlieger, Jun 06 2019, after Giovanni Resta at A034448 and Amiram Eldar at A048250. *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325973(n) = ((A034448(n)+A048250(n))/2);

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));

a(n) = (1/2) * (A034448(n) + A048250(n)).
a(n) = A000203(n) - A325974(n).
a(n) = n + A325977(n).
a(A048107(n)) = A326043(A048107(n)).
For n >= 1, a(2^n) = A052548(n-1) = 2^(n-1) + 2.
For n >= 1, a(3^n) = A289521(n) = (3^n + 5)/2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) + 1)/4 = 0.5921081944... . - Amiram Eldar, Feb 22 2024

A036690 Product of a prime and the following number.

Original entry on oeis.org

6, 12, 30, 56, 132, 182, 306, 380, 552, 870, 992, 1406, 1722, 1892, 2256, 2862, 3540, 3782, 4556, 5112, 5402, 6320, 6972, 8010, 9506, 10302, 10712, 11556, 11990, 12882, 16256, 17292, 18906, 19460, 22350, 22952, 24806, 26732, 28056, 30102

Offset: 1

Felice Russo

The infinite sum over the reciprocals is given in A179119. - Wolfdieter Lang, Jul 10 2019

1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is positive and even. - Amiram Eldar, Jan 23 2021

			a(3)=30 because prime(3)=5 and prime(3)+1=6, hence 5*6 = 30.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A036689, A034953, A065463, A179119, A306633.

[p^2+p: p in PrimesUpTo(250)]; // Vincenzo Librandi, Dec 19 2010

Table[(Prime[n] + 1) Prime[n], {n, 1, 100}] (* Artur Jasinski, Feb 06 2007 *)

a(n)=my(p=prime(n)); p*(p+1) \\ Charles R Greathouse IV, Mar 27 2020

a(n) = prime(n)*(prime(n)+1).
a(n) = A060800(n) - 1.
a(n) = 2*A034953(n). - Artur Jasinski, Feb 06 2007
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(3) (A306633).
Product_{n>=1} (1 - 1/a(n)) = A065463. (End)
Sum_{n>=1} 1/a(n) = A179119. - R. J. Mathar, Mar 31 2025

A325977 a(n) = (1/2)*(A034460(n) + A325313(n)).

Original entry on oeis.org

0, 1, 1, 0, 1, 6, 1, -2, -2, 8, 1, 4, 1, 10, 9, -6, 1, 3, 1, 4, 11, 14, 1, 0, -9, 16, -11, 4, 1, 42, 1, -14, 15, 20, 13, -5, 1, 22, 17, -4, 1, 54, 1, 4, -3, 26, 1, -8, -20, -2, 21, 4, 1, -6, 17, -8, 23, 32, 1, 36, 1, 34, -7, -30, 19, 78, 1, 4, 27, 74, 1, -21, 1, 40, -11, 4, 19, 90, 1, -20, -38, 44, 1, 44, 23, 46, 33, -16, 1, 36, 21, 4

Offset: 1

Antti Karttunen, Jun 02 2019

sign

Question: Are n = 1, 4, 24, 240, 349440 (A325963) the only positions of zeros in this sequence?

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A033879, A034448, A034460, A048250, A306633, A325313, A325963, A325973, A325974, A325975, A325978, A325979, A325981.

Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #2) - 2 #1 + Times @@ (1 + #2[[;; , 1]]) & @@ {#, FactorInteger[#]}] &, 90] (* Michael De Vlieger, Jun 06 2019, after Giovanni Resta at A034448 and Amiram Eldar at A048250. *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);

a(n) = (1/2)*(A034460(n) + A325313(n)).
a(n) = A325973(n) - n.
a(n) = A325978(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/4 = 0.0921081944... . - Amiram Eldar, Feb 22 2024

A371242 The sum of the unitary divisors of n that are cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 1, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 3, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96

Offset: 1

Amiram Eldar, Mar 16 2024

The number of these divisors is A365498(n).

D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Francesco Pappalardi, A survey on k-freeness, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.

Cf. A000203, A004709, A005117, A034448, A036966, A065465, A092261 (squarefree analog), A306633, A365498.

f[p_, e_] := If[e < 3, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 3, f[i, 1]^f[i, 2] + 1, 1));}

Multiplicative with a(p^e) = p^e + 1 for e <= 2, and a(p^e) = 1 for e >= 3.
a(n) = 1 if and only if n is cubefull (A036966).
a(n) = A000203(n) if and only if n is squarefree (A005117).
a(n) <= A034448(n), with equality if and only if n is cubefree.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) - 1/p^(3*s-2)).
Sum_{j=1..n} a(j) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^3 + 1/(p^2 + p)) = 1.16545286600957717104.... .
In general, the formula above holds for the sum of unitary divisors of n that are k-free numbers (k >= 2) with c = Product_{p prime} (1 - 1/p^k + 1/(p^2 + p)) (Suryanarayana, 1969). If k = 2 then c = A065465. In the limit when k -> oo, c = A306633.

A107749 OrdinaryUnitarySigma(n): If n = Product p_i^r_i then OUSigma(n) = Sigma(2^r_1)*UnitarySigma(n/2^r_1).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 10, 18, 12, 28, 14, 24, 24, 31, 18, 30, 20, 42, 32, 36, 24, 60, 26, 42, 28, 56, 30, 72, 32, 63, 48, 54, 48, 70, 38, 60, 56, 90, 42, 96, 44, 84, 60, 72, 48, 124, 50, 78, 72, 98, 54, 84, 72, 120, 80, 90, 60, 168, 62, 96, 80, 127, 84, 144, 68, 126, 96

Offset: 1

Yasutoshi Kohmoto, Jun 11 2005, Feb 24 2007

The sum of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079). - Amiram Eldar, Aug 26 2023

			OUSigma(2^4*7^2) = Sigma(2^4)*UnitarySigma(7^2) = 31*50 = 1550.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000079, A069184, A091321, A000203, A034448, A006519, A306633.

A107749 := proc(n) local a,f,p,e; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; if p = 2 then a := a*(2^(e+1)-1) ; else a := a*(p^e+1) ; end if; end do; a ; end proc: # R. J. Mathar, Jun 02 2011

f[2, e_] := 2^(e+1)-1; f[p_, e_] := p^e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],if(fm[k,1]==2,2^(fm[k,2]+1)-1,fm[k,1]^fm[k,2]+1))

a(n) = A000203(p2) * A034448(n/p2), where p2 = A006519(n). - R. J. Mathar, Jun 15 2008
Multiplicative with a(2^e) = 2^(e+1)-1, a(p^e) = p^e+1 for p>2, e>0.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/7) * zeta(2)/zeta(3) = (4/7) * A306633 = 0.781961... . - Amiram Eldar, Nov 01 2022
Dirichlet g.f.: (4^s/(4^s-2)) * zeta(s)*zeta(s-1)/zeta(2*s-1). - Amiram Eldar, Aug 26 2023

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007
More terms from R. J. Mathar, Jun 15 2008
Name corrected by Franklin T. Adams-Watters, Aug 24 2013

cp's OEIS Frontend

A000010 Euler totient function phi(n): count numbers <= n and prime to n.

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A065463 Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))).

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A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

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A104141 Decimal expansion of 3/Pi^2.

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A046125 Number of negative fundamental discriminants having class number n.

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A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

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