A023896 -id:A023896 - cp's OEIS frontend

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613

Offset: 1

N. J. A. Sloane, Mira Bernstein and Robert G. Wilson v

Also, odd primes p such that -3 is a square mod p. - N. J. A. Sloane, Dec 25 2017
Equivalently, primes of the form p = (x^3 - y^3)/(x - y). If x=y+1 we get the cuban primes A002407, which is therefore a subsequence.
These are not to be confused with the Eisenstein primes, which are the primes in the ring of integers Z[w], where w = (-1+sqrt(-3))/2. The present sequence gives the rational primes which are also Eisenstein primes. - N. J. A. Sloane, Feb 06 2008
Also primes of the form x^2+3y^2 and, except for 3, x^2+xy+7y^2. See A140633. - T. D. Noe, May 19 2008
Conjecture: this sequence is Union(A002383,A162471). - Daniel Tisdale, Jul 04 2009
Primes p such that antiharmonic mean B(p) of the numbers k < p such that gcd(k, p) = 1 is not integer, where B(p) = A053818(p) / A023896(p) = A175505(p) / A175506(p) = (2p - 1) / 3. Primes p such that A175506(p) > 1. Subsequence of A179872. Union a(n) + A179891 = A179872. Example: a(6) = 37 because B(37) = A053818(37) / A023896(37) = A175505(37) / A175506(37) = 16206 / 666 = 73 / 3 (not integer). Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Subsequence of Loeschian numbers, cf. A003136 and A024614; A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011
Primes such that there exist a unique x, y, with 1 < x <= y < p, x + y == 1 (mod p) and x * y == 1 (mod p). - Jon Perry, Feb 02 2014
The prime factors of A002061. - Richard R. Forberg, Dec 10 2014
This sequence gives the primes p which solve s^2 == -3 (mod 4*p) (see Buell, Proposition 4.1., p. 50, for Delta = -3). p = 2 is not a solution. x^2 == -3 (mod 4) has solutions for all odd x. x^2 == -3 (mod  p) has for odd primes p, not 3, the solutions of Legendre(-3|p) = +1 which are p == {1, 7} (mod 12). For p = 3 the representative solution is x = 0. Hence the solution of s^2 == -3 (mod 4*p) are the odd primes p = 3 and p == {1, 7} (mod 12) (or the primes p = 0, 1 (mod 3)). - Wolfdieter Lang, May 22 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 50.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

T. D. Noe, Table of n, a(n) for n = 1..1000
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Eisenstein Integer.

Subsequence of A003136.
Subsequences include A002407, A002648, and A201477.
Apart from initial term, same as A045331.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2).
Cf. A000040, A003627.
Primes in A003136 and A034017.

a007645 n = a007645_list !! (n-1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
-- Reinhard Zumkeller, Jul 11 2013, Oct 30 2011

select(isprime,[3, seq(6*k+1, k=1..1000)]); # Robert Israel, Dec 12 2014

Join[{3},Select[Prime[Range[150]],Mod[#,3]==1&]] (* Harvey P. Dale, Aug 21 2021 *)

forprime(p=2,1e3,if(p%3<2,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

p == 0 or 1 (mod 3).
{3} UNION A002476. - R. J. Mathar, Oct 28 2008
A007645 UNION A003627 = A000040. - Juri-Stepan Gerasimov, Jan 28 2010

Entry revised by N. J. A. Sloane, Jan 29 2013

A023022 Number of partitions of n into two relatively prime parts. After initial term, this is the "half-totient" function phi(n)/2 (A000010(n)/2).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 3, 2, 5, 2, 6, 3, 4, 4, 8, 3, 9, 4, 6, 5, 11, 4, 10, 6, 9, 6, 14, 4, 15, 8, 10, 8, 12, 6, 18, 9, 12, 8, 20, 6, 21, 10, 12, 11, 23, 8, 21, 10, 16, 12, 26, 9, 20, 12, 18, 14, 29, 8, 30, 15, 18, 16, 24, 10, 33, 16, 22, 12, 35, 12, 36, 18, 20, 18, 30, 12, 39, 16, 27, 20, 41, 12

Offset: 2

N. J. A. Sloane

The number of distinct linear fractional transformations of order n. Also the half-totient function can be used to construct a tree containing all the integers. On the zeroth rank we have just the integers 1 and 2: immediate "ancestors" of 1 and 2 are (1: 3,4,6 2: 5,8,10,12) etc. - Benoit Cloitre, Jun 03 2002
Moebius transform of floor(n/2). - Paul Barry, Mar 20 2005
Also number of different kinds of regular n-gons, one convex, the others self-intersecting. - Reinhard Zumkeller, Aug 20 2005
From Artur Jasinski, Oct 28 2008: (Start)
Degrees of minimal polynomials of cos(2*Pi/n). The first few are
   1: x - 1
   2: x + 1
   3: 2*x + 1
   4: x
   5: 4*x^2 + 2*x - 1
   6: 2*x - 1
   7: 8*x^3 + 4*x^2 - 4*x - 1
   8: 2*x^2 - 1
   9: 8*x^3 - 6*x + 1
  10: 4*x^2 - 2*x - 1
  11: 32*x^5 + 16*x^4 - 32*x^3 - 12*x^2 + 6*x + 1
These polynomials have solvable Galois groups, so their roots can be expressed by radicals. (End)
a(n) is the number of rationals p/q in the interval [0,1] such that p + q = n. - Geoffrey Critzer, Oct 10 2011
It appears that, for n > 2, a(n) = A023896(n)/n. Also, it appears that a record occurs at n > 2 in this sequence if and only if n is a prime.  For example, records occur at n=5, 7, 11, 13, 17, ..., all of which are prime. - John W. Layman, Mar 26 2012
From Wolfdieter Lang, Dec 19 2013: (Start)
a(n) is the degree of the algebraic number of s(n)^2 = (2*sin(Pi/n))^2, starting at a(1)=1. s(n) = 2*sin(Pi/n) is the length ratio side/R for a regular n-gon inscribed in a circle of radius R (in some length units). For the coefficient table of the minimal polynomials of s(n)^2 see A232633.
Because for even n, s(n)^2 lives in the algebraic number field Q(rho(n/2)), with rho(k) = 2*cos(Pi/k), the degree is a(2*l) = A055034(l). For odd n, s(n)^2 is an integer in Q(rho(n)), and the degree is a(2*l+1) = A055034(2*l+1) = phi(2*l+1)/2, l >= 1, with Euler's totient phi=A000010 and a(1)=1. See also A232631-A232633.
(End)
Also for n > 2: number of fractions A182972(k)/A182973(k) such that A182972(k) + A182973(k) = n, A182972(n) and A182973(n) provide an enumeration of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator. - Reinhard Zumkeller, Jul 30 2014
Number of distinct rectangles with relatively prime length and width such that L + W = n, W <= L. For a(17)=8; the rectangles are 1 X 16, 2 X 15, 3 X 14, 4 X 13, 5 X 12, 6 X 11, 7 X 10, 8 X 9. - Wesley Ivan Hurt, Nov 12 2017
After including a(1) = 1, the number of elements of any reduced residue system mod* n used by Brändli and Beyne is a(n). See the examples below. - Wolfdieter Lang, Apr 22 2020
a(n) is the number of ABC triples with n = c. - Felix Huber, Oct 12 2023

			a(15)=4 because there are 4 partitions of 15 into two parts that are relatively prime: 14 + 1, 13 + 2, 11 + 4, 8 + 7. - _Geoffrey Critzer_, Jan 25 2015
The smallest nonnegative reduced residue system mod*(n) for n = 1 is {0}, hence a(1) = 1; for n = 9 it is {1, 2, 4}, because 5 == 4 (mod* 9) since -5 == 4 (mod 9), 7 == 2 (mod* 9) and 8 == 1 (mod* 9). Hence a(9) = phi(9)/2 = 3. See the comment on Brändli and Beyne above. - _Wolfdieter Lang_, Apr 22 2020

G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Sect. 6, Problems 60&61.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 2..10000
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.
K. S. Brown, The Half-Totient Tree
Tianxin Cai, Zhongyan Shen, and Mengjun Hu, On the Parity of the Generalized Euler Function, Advances in Mathematics (China), 2013, 42(4): 505-510.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 518.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Pinthira Tangsupphathawat, Takao Komatsu, and Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
Eric Weisstein's World of Mathematics, Polygon Triangle Picking
Eric Weisstein's World of Mathematics, Trigonometry Angles
Canze Zhu and Qunying Liao, A recursion formula for the generalized Euler function phi_e(n), arXiv:2105.10870 [math.NT], 2021.

Cf. A000010, A055684, A046657, A049806, A049703, A062956.
Cf. A181875, A181876, A181877, A183918.
Cf. A023896.
Cf. A182972, A182973, A245497, A245718.

a023022 n = length [(u, v) | u <- [1 .. div n 2],
                             let v = n - u, gcd u v == 1]
-- Reinhard Zumkeller, Jul 30 2014

[1] cat [EulerPhi(n)/ 2: n in [3..100]]; // Vincenzo Librandi, Aug 19 2018

A023022 := proc(n)
    if n =2 then
        1;
    else
        numtheory[phi](n)/2 ;
    end if;
end proc:
seq(A023022(n),n=2..60) ; # R. J. Mathar, Sep 19 2017

Join[{1}, Table[EulerPhi[n]/2, {n, 3, 100}]] (* adapted by Vincenzo Librandi, Aug 19 2018 *)

a(n)=if(n<=2,1,eulerphi(n)/2);
/* for printing minimal polynomials of cos(2*Pi/n) */
default(realprecision,110);
for(n=1,33,print(n,": ",algdep(cos(2*Pi/n),a(n))));

from sympy.ntheory import totient
def a(n): return 1 if n<3 else totient(n)/2 # Indranil Ghosh, Mar 30 2017

a(n) = phi(n)/2 for n >= 3.
a(n) = (1/n)*Sum_{k=1..n-1, gcd(n, k)=1} k = A023896(n)/n for n>2. - Reinhard Zumkeller, Aug 20 2005
G.f.: x*(x - 1)/2 + (1/2)*Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Apr 13 2017
a(n) = Sum_{d|n} moebius(n/d)*floor(d/2). - Michel Marcus, May 25 2021

This was in the 1973 "Handbook", but then was dropped from the database. Resubmitted by David W. Wilson
Entry revised by N. J. A. Sloane, Jun 10 2012
Polynomials edited with the consent of Artur Jasinski by Wolfdieter Lang, Jan 08 2011
Name clarified by Geoffrey Critzer, Jan 25 2015

A001783 n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.

Original entry on oeis.org

1, 1, 2, 3, 24, 5, 720, 105, 2240, 189, 3628800, 385, 479001600, 19305, 896896, 2027025, 20922789888000, 85085, 6402373705728000, 8729721, 47297536000, 1249937325, 1124000727777607680000, 37182145, 41363226782215962624, 608142583125, 1524503639859200000

Offset: 1

N. J. A. Sloane

In other words, a(1) = 1 and for n >= 2, a(n) = product of the phi(n) numbers < n and relatively prime to n.
From Gauss's generalization of Wilson's theorem (see Weisstein link) it follows that, for n>1, a(n) == -1 (mod n) if and only if there exists a primitive root modulo n (cf. the Hardy and Wright reference, Theorem 129. p. 102). - Vladimir Shevelev, May 11 2012
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 25 2016
Cosgrave & Dilcher propose the name Gauss factorial. Indeed the sequence is the special case N = n of the Gauss factorial N_n! = Product_{1<=j<=N, gcd(j, n)=1} j (see A216919). - Peter Luschny, Feb 07 2018

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 129, p. 102.
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).

T. D. Noe, Table of n, a(n) for n = 1..200
J. B. Cosgrave and K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008)
J. B. Cosgrave and K. Dilcher, The multiplicative orders of certain Gauss factorials, Intl. J. Number Theory 7 (1) (2011) 145-171.
S. W. Golomb and William Small, Problem E1045, Amer. Math. Monthly, 60 (1953), 422.
Muhammed H. Islam and Shahriar Manzoor, φ1 and phitorial are injections, for any positive integer N, where N > 1
Laszlo Toth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7
Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A023896, A066570, A038566, A124441.

Main diagonal gives A216919.

a001783 = product . a038566_row
-- Reinhard Zumkeller, Mar 04 2012, Aug 26 2011

A001783 := proc(n) local i,t1; t1 := 1; for i from 1 to n do if gcd(i,n)=1 then t1 := t1*i; fi; od; t1; end;
A001783 := proc(n) local i; mul(i,i=select(k->igcd(n,k)=1,[$1..n])) end; # Peter Luschny, Oct 30 2010

A001783[n_]:=Times@@Select[Range[n],CoprimeQ[n,#]&];
Array[A001783,20] (* Enrique Pérez Herrero, Jul 23 2011 *)

A001783(n)=prod(k=2,n-1,k^(gcd(k,n)==1))  \\ M. F. Hasler, Jul 23 2011

a(n)=my(f=factor(n),t=n^eulerphi(f)); fordiv(f,d, t*=(d!/d^d)^moebius(n/d)); t \\ Charles R Greathouse IV, Nov 05 2015

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A001783(n): return Gauss_factorial(n, n)
[A001783(n) for n in (1..25)] # Peter Luschny, Oct 01 2012

a(n) = n^phi(n)*Product_{d|n} (d!/d^d)^mu(n/d); phi=A000010 is the Euler totient function and mu=A008683 the Moebius function (Tom M. Apostol, Introduction to Analytic Number Theory, New York 1984, p. 48). - Franz Vrabec, Jul 08 2005
a(n) = n!/A066570(n). - R. J. Mathar, Mar 10 2011
A001221(a(n)) = A000720(n) - A001221(n) = A048865(n).
A006530(a(n)) = A136548(n). - Enrique Pérez Herrero, Jul 23 2011
a(n) = A124441(n)*A124442(n). - M. F. Hasler, Jul 23 2011
a(n) == (-1)^A211487(n) (mod n). - Vladimir Shevelev, May 13 2012
a(n) = A250269(n) / A193679(n). - Daniel Suteu, Apr 05 2021

More terms from James Sellers, Dec 23 1999

A053818 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^2.

Original entry on oeis.org

1, 1, 5, 10, 30, 26, 91, 84, 159, 140, 385, 196, 650, 406, 620, 680, 1496, 654, 2109, 1080, 1806, 1650, 3795, 1544, 4150, 2756, 4365, 3164, 7714, 2360, 9455, 5456, 7370, 6256, 9940, 5196, 16206, 8778, 12324, 8560, 22140, 6972, 25585

Offset: 1

N. J. A. Sloane, Apr 07 2000

nonn

Equals row sums of triangle A143612. - Gary W. Adamson, Aug 27 2008
a(n) = A175505(n) * A023896(n) / A175506(n). For number n >= 1 holds B(n) = a(n) / A023896(n) = A175505(n) / A175506(n), where B(n) = antiharmonic mean of numbers k such that GCD(k, n) = 1 for k < n. - Jaroslav Krizek, Aug 01 2010
n does not divide a(n) iff n is a term in A316860, that is, either n is a power of 2 or n is a multiple of 3 and no prime factor of n is congruent to 1 mod 3. - Jianing Song, Jul 16 2018

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_2(n).
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #2.

G. C. Greubel, Table of n, a(n) for n = 1..2500
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
Geoffrey B. Campbell, Dirichlet summations and products over primes, Int. J. Math. Math. Sci. 16 92) (1993) 359. eq. (3.1)
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], 2016-2018.

Cf. A023896, A053819, A053820.
Cf. A143612. - Gary W. Adamson, Aug 27 2008
Cf. A179871-A179880, A179882-A179887, A179890, A179891, A007645, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Column k=2 of A308477.

A053818 := proc(n)
    local a,k;
    a := 0 ;
    for k from 1 to n do
        if igcd(k,n) = 1 then
            a := a+k^2 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 26 2013

a[n_] := Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2); Array[a, 43] (* Robert G. Wilson v, Jul 01 2010 *)
a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^2/3) * Times @@ ((p - 1)*p^(e - 1)) + (n/6) * Times @@ (1 - p)]; Array[a, 100] (* Amiram Eldar, Dec 03 2023 *)

a(n) = sum(k=1, n, k^2*(gcd(n,k) == 1)); \\ Michel Marcus, Jan 30 2016

a(n) = {my(f = factor(n)); if(n == 1, 1, (n^2/3) * eulerphi(f) + (n/6) * prod(i = 1, #f~, 1 - f[i, 1]));} \\ Amiram Eldar, Dec 03 2023

If n = p_1^e_1 * ... *p_r^e_r then a(n) = n^2*phi(n)/3 + (-1)^r*p_1*..._p_r*phi(n)/6.
a(n) = n^2*A000010(n)/3 + n*A023900(n)/6, n>1. [Brown]
a(n) = (A000010(n)/3) * (n^2 + (-1)^A001221(n)*A007947(n)/2) for n>=2. - Jaroslav Krizek, Aug 24 2010
G.f. A(x) satisfies: A(x) = x*(1 + x)/(1 - x)^4 - Sum_{k>=2} k^2 * A(x^k). - Ilya Gutkovskiy, Mar 29 2020
Sum_{k=1..n} a(k) ~ n^4 / (2*Pi^2). - Amiram Eldar, Dec 03 2023

A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

Original entry on oeis.org

1, 2, 4, 6, 11, 11, 22, 22, 31, 32, 56, 39, 79, 65, 74, 86, 137, 92, 172, 116, 151, 167, 254, 151, 261, 236, 274, 237, 407, 221, 466, 342, 389, 410, 452, 336, 667, 515, 550, 452, 821, 452, 904, 611, 641, 761, 1082, 599, 1051, 782, 956, 864, 1379, 821, 1166

Offset: 1

Henry Gould, Oct 15 2000

Sum of numerators of n-th order Farey series (cf. A006842). - Benoit Cloitre, Oct 28 2002
Equals row sums of triangle A143613. - Gary W. Adamson, Aug 27 2008
Equals row sums of triangle A159936. - Gary W. Adamson, Apr 26 2009
Also row sums of triangle A164306. - Reinhard Zumkeller, Aug 12 2009

H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), 39 (1997), 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), 39 (1997), 183-194.

T. D. Noe, Table of n, a(n) for n = 1..1000
Zachary Franco, Problem 12114, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 469; A Dirichlet Series with Reduced Numerators, Solution to Problem 12114 by Tamas Wiandt, ibid., Vol. 128, No. 1 (2021), pp. 91-92.
Index entries for sequences related to lcm's.

Cf. A000010, A000217, A006842, A018804, A023896, A051193, A057660, A143613, A159936, A164306.

See A341316 for another version.

a057661 n = a051193 n `div` n  -- Reinhard Zumkeller, Jun 10 2015

[&+[&+[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(n)]: n in [1..100]]; // Jaroslav Krizek, Dec 28 2016

Table[Total[Numerator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *)

a(n)=sum(k=1,n,lcm(n,k))/n \\ Charles R Greathouse IV, Feb 07 2017

from math import lcm
def A057661(n): return sum(lcm(n,k)//n for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057661(n): return 1+prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items())>>1 # Chai Wah Wu, Aug 05 2024

a(n) = (1+A057660(n))/2.
a(n) = A051193(n)/n.
a(n) = Sum_{d|n} psi(d), where psi(m) = is the sum of totatives of m (A023896). - Jaroslav Krizek, Dec 28 2016
a(n) = Sum_{i=1..n} denominator(n/i). - Wesley Ivan Hurt, Feb 26 2017
G.f.: x/(2*(1 - x)) + (1/2)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017
If p is prime, then a(p) = T(p-1) + 1 = p(p-1)/2 + 1, where T(n) = n(n+1)/2 is the n-th triangular number (A000217). - David Terr, Feb 10 2019
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, May 29 2021
Dirichlet g.f.: zeta(s)*(1 + zeta(s-2)/zeta(s-1))/2 (Franco, 2019). - Amiram Eldar, Mar 26 2022

More terms from James Sellers, Oct 16 2000

A076512 Denominator of cototient(n)/totient(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 2, 10, 1, 12, 3, 8, 1, 16, 1, 18, 2, 4, 5, 22, 1, 4, 6, 2, 3, 28, 4, 30, 1, 20, 8, 24, 1, 36, 9, 8, 2, 40, 2, 42, 5, 8, 11, 46, 1, 6, 2, 32, 6, 52, 1, 8, 3, 12, 14, 58, 4, 60, 15, 4, 1, 48, 10, 66, 8, 44, 12, 70, 1, 72, 18, 8, 9, 60, 4, 78, 2, 2, 20, 82, 2, 64, 21

Offset: 1

Reinhard Zumkeller, Oct 15 2002

a(n)=1 iff n=A007694(k) for some k.
Numerator of phi(n)/n=Prod_{p|n} (1-1/p). - Franz Vrabec, Aug 26 2005
From Wolfdieter Lang, May 12 2011: (Start)
For n>=2, a(n)/A109395(n) = sum(((-1)^r)*sigma_r,r=0..M(n)) with the elementary symmetric functions (polynomials) sigma_r of the indeterminates {1/p_1,...,1/p_M(n)} if n = prod((p_j)^e(j),j=1..M(n)) where M(n)=A001221(n) and sigma_0=1.
This follows by expanding the above given product for phi(n)/n.
The n-th member of this rational sequence 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5,... is also (2/n^2)*sum(k,with 1<=k=2.
Therefore, this scaled sum depends only on the distinct prime factors of n.
See also A023896. Proof via PIE (principle of inclusion and exclusion). (End)
In the sequence of rationals r(n)=eulerphi(n)/n: 1, 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5, 10/11, 1/3, ... one can observe that new values are obtained for squarefree indices (A005117); while for a nonsquarefree number n (A013929), r(n) = r(A007947(n)), where A007947(n) is the squarefree kernel of n. - Michel Marcus, Jul 04 2015

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A076511 (numerator of cototient(n)/totient(n)), A051953.

Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).

[Numerator(EulerPhi(n)/n): n in [1..100]]; // Vincenzo Librandi, Jul 04 2015

Table[Denominator[(n - EulerPhi[n])/EulerPhi[n]], {n, 80}] (* Alonso del Arte, May 12 2011 *)

vector(80, n, numerator(eulerphi(n)/n)) \\ Michel Marcus, Jul 04 2015

a(n) = A000010(n)/A009195(n).

A179871 Numbers h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is an integer.

Original entry on oeis.org

1, 2, 5, 10, 11, 17, 22, 23, 29, 34, 41, 46, 47, 53, 55, 58, 59, 71, 82, 83, 85, 89, 91, 94, 101, 106, 107, 110, 113, 115, 118, 131, 133, 137, 142, 145, 149, 166, 167, 170, 173, 178, 179, 182, 187, 191, 197, 202, 205, 214

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Numbers h such that B(h) = A053818(h) / A023896(h) = A175505(h) / A175506(h) is an integer.
Numbers h such that A175506(h) = 1.
Complement of A179872.
See A179873 (odd positive integers) for corresponding values A175505(a(n)).
Union of A003627 (primes of form 3n-1) and A179887.

			a(9) = 29 because B(29) = A053818(29) / A023896(29) = 7714/406 = 19 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Ivan Neretin)

Cf. A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

B[n_] := Plus @@ ((ks = Select[Range[n], GCD[n, #] == 1 &])^2)/Plus @@ ks; Select[Range[215], IntegerQ[B[#]] &] (* Ivan Neretin, May 22 2015 *)

isok(k) = {my(f = factor(k)); if(k == 1, 1, denominator(2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)) == 1);} \\ Amiram Eldar, May 24 2025

A179891 Composites h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is not integer.

Original entry on oeis.org

4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 56, 57, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 111

Offset: 1

Jaroslav Krizek, Jul 30 2010

nonn

Composites h such that B(h) = A053818(h) / A023896(h) = A175505(h) / A175506(h) is not integer.
Composites h such that A175506(h) > 1.
Subsequence of A179872.
A179872 is the union of this sequence and A007645.

			a(6) = 14 because B(14) = A053818(14) / A023896(14) = 406/42 = 29/3 (not integer).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1749 from G. C. Greubel)

Cf. A007645, A023896, A053818, A175505, A175506.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890.

f[n_] := 2 Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2)/(n*EulerPhi@n); Select[ Range@ 111, ! PrimeQ@# && ! IntegerQ@f@# &] (* Robert G. Wilson v, Aug 02 2010 *)

isok(k) = if(isprime(k), 0, my(f = factor(k)); if(k == 1, 0, denominator(2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)) > 1)); \\ Amiram Eldar, May 25 2025

More terms from Robert G. Wilson v, Aug 02 2010

A179875 Numbers h such that h and h+1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1.

Original entry on oeis.org

1, 6, 10, 22, 46, 58, 65, 69, 77, 82, 106, 129, 166, 178, 185, 194, 210, 221, 226, 237, 254, 262, 265, 309, 321, 330, 346, 358, 365, 382, 398, 417, 437, 454, 462, 466, 469, 473, 478, 482, 493, 497, 502

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Corresponding values of numbers h+1 see A179876.
Numbers h such that A175505(h) = A175505(h+1).
numbers h such that A175506(h) = A175506(h+1).
Antiharmonic mean B(h) of numbers k such that gcd(k, h) = 1 for numbers h >= 1 and k < h = A053818(n) / A023896(n) = A175505(h) / A175506(h).
Conjecture: also numbers k such that mu(k) = 1 and mu(k+1) = -1, where mu is the Möbius function (tested on the first 10^4 terms). - Amiram Eldar, Mar 06 2021

			For n=3: a(3) = 10; B(10) = A175505(10) / A175506(10) = 7, B(11) = A175505(11) / A175506(11) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A179871, A179872, A179873, A179874, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

A179876 Numbers h such that h and h-1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1.

Original entry on oeis.org

2, 7, 11, 23, 47, 59, 66, 70, 78, 83, 107, 130, 167, 179, 186, 195, 211, 222, 227, 238, 255, 263, 266, 310, 322, 331, 347, 359, 366, 383, 399, 418, 438, 455, 463, 467, 470, 474, 479, 483, 494, 498, 503

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Corresponding values of numbers h-1 see A179875.
Numbers h such that A175505(h) = A175505(h-1).
Numbers h such that A175506(h) = A175506(h-1).
Antiharmonic mean B(h) of numbers k such that gcd(k, h) = 1 for numbers h >= 1 and k < h = A053818(n) / A023896(n) = A175505(h) / A175506(h).

			For n=3: a(3) = 11; B(11) = A175505(11) / A175506(11) = 7, B(10) = A175505(10) / A175506(10) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2047 from R. J. Mathar)

Cf. A179871-A179880, A179882-A179887, A179890, A179891.

antiHMeanGcd := proc(h)
        option remember;
        local a023896,a053818,k ;
        a023896 := 0 ;
        a053818 := 0 ;
        for k from 1 to h do
                if igcd(k,h) = 1 then
                        a023896 := a023896+k ;
                        a053818 := a053818+k^2 ;
                end if;
        end do:
        a053818/a023896 ;
end proc:
n := 1:
for h from 2 do
        if antiHMeanGcd(h) = antiHMeanGcd(h-1) then
                printf("%d %d\n",n,h) ;
                n := n+1 ;
        end if;
end do: # R. J. Mathar, Sep 26 2013

hmax = 1000;
antiHMeanGcd[h_] := antiHMeanGcd[h] = Module[{num = 0, den = 0, k}, For[k = 1, k <= h, k++, If[GCD[k, h] == 1, den += k; num += k^2]]; num/den];
Reap[n = 1; For[h = 2, h <= hmax, h++, If[antiHMeanGcd[h] == antiHMeanGcd[h - 1], Sow[h]; n++]]][[2, 1]] (* Jean-François Alcover, Mar 23 2020, after R. J. Mathar *)

cp's OEIS Frontend

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

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A023022 Number of partitions of n into two relatively prime parts. After initial term, this is the "half-totient" function phi(n)/2 (A000010(n)/2).

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A001783 n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.

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A053818 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^2.

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A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

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