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

A007434 Jordan function J_2(n) (a generalization of phi(n)).

Original entry on oeis.org

1, 3, 8, 12, 24, 24, 48, 48, 72, 72, 120, 96, 168, 144, 192, 192, 288, 216, 360, 288, 384, 360, 528, 384, 600, 504, 648, 576, 840, 576, 960, 768, 960, 864, 1152, 864, 1368, 1080, 1344, 1152, 1680, 1152, 1848, 1440, 1728, 1584, 2208, 1536
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Number of points in the bicyclic group Z/mZ X Z/mZ whose order is exactly m. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Mar 14 2006
Number of irreducible fractions among {(u+v*i)/n : 1 <= u, v <= n} with i = sqrt(-1), where a fraction (u+v*i)/n is called irreducible if and only if gcd(u, v, n) = 1. - Reinhard Zumkeller, Aug 20 2005
The weight of the n-th polynomial for the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, let the weight of b1 = 1, b2 = 3, b3 = 8, b4 = 12 and let e1 = b1, e2 = b2*b1, e3 = b3*b1, e4 = b2*b4*b1, e5 = (b2^4*b4 - b3^3)*b1 = b5*e1, and so on, be an elliptic divisibility sequence. Then weight of e2 = 4, e3 = 9, e4 = 16, e5 = 25, where weight of en is n^2 in general, while weight of bn is a(n). - Michael Somos, Aug 12 2008
J_2(n) divides J_{2k}(n). J_2(n) gives the number of 2-tuples (x1,x2), such that 1 <= x1, x2 <= n and gcd(x1, x2, n) = 1. - Enrique Pérez Herrero, Mar 05 2011
From Jianing Song, Apr 06 2019: (Start)
Let k be any quadratic field such that all prime factors of n are inert in k, O_k be the corresponding ring of integers and G(n) = (O_k/nO_k)* be the multiplicative group of integers in O_k modulo n, then a(n) is the number of elements in G(n). The exponent of G(n) is A306933(n). [Equivalently, G(p^e) can be defined as (Z_{p^2}/p^eZ_{p^2})*, where Z_{p^2} is the ring of integers of the field Q_{p^2} (with a unique maximal ideal pZ_{p^2}), and Q_{p^2} is the unique unramified quadratic extension of the p-adic field Q_p. For the group structure of G(p^e), see A306933. - Jianing Song, Jun 19 2025]
For n >= 5, a(n) is divisible by 24. (End)
The Del Centina article on page 106 mentions a formula by Halphen denoted by phi(n)T(n). - Michael Somos, Feb 05 2021

Examples

			a(4) = 12 because the divisors of 4 being 1, 2, 4, we find that phi(1)*phi(4/1)*(4/1) = 8, phi(2)*phi(4/2)*(4/2) = 2, phi(4)*phi(4/4)*(4/4) = 2 and 8 + 2 + 2 = 12.
G.f. = x + 3*x^2 + 8*x^3 + 12*x^4 + 24*x^5 + 24*x^6 + 48*x^7 + 48*x^8 + 72*x^9 + ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
  • A. Del Centina, Poncelet's porism: a long story of renewed discoveries, I, Hist. Exact Sci. (2016), v. 70, p. 106.
  • L. E. Dickson (1919, repr. 1971). History of the Theory of Numbers I. Chelsea. p. 147.
  • P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.
  • G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Section 6, Problem 64.
  • M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. 206. Springer-Verlag. p. 11.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000056, A000290, A001065, A001615, A027750, A046970, A115000, A134675.
Cf. A059379 and A059380 (triangle of values of J_k(n)).
Cf. A000010 (J_1), this sequence (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A002117, A088453, A301875, A301876, A321879 (partial sums).

Programs

  • Haskell
    a007434 n = sum $ zipWith3 (\x y z -> x * y * z)
                  tdivs (reverse tdivs) (reverse divs)
                  where divs = a027750_row n;  tdivs = map a000010 divs
-- Reinhard Zumkeller, Nov 24 2012
  • Maple
    J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 2)
A007434 := proc(n)
    add(d^2*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Nov 03 2015
  • Mathematica
    jordanTotient[n_, k_:1] := DivisorSum[n, #^k*MoebiusMu[n/#] &] /; (n > 0) && IntegerQ[n]; Table[jordanTotient[n, 2], {n, 48}] (* Enrique Pérez Herrero, Sep 14 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ n/d], {d, Divisors @ n}]]; (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], n^2 (Times @@ ((1 - 1/#[[1]]^2) & /@ FactorInteger @ n))]; (* Michael Somos, Jan 11 2014 *)
jordanTotient[n_Integer?Positive, r_:1] := DirichletConvolve[MoebiusMu[K], K^r, K, n]; Table[jordanTotient[n, 2], {n, 48}] (* Jan Mangaldan, Jun 03 2016 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * moebius(n / d)))}; /* Michael Somos, Mar 20 2004 */
  • PARI
    {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X) / (1 - X*p^2))[n])}; /* Michael Somos, Jan 11 2014 */
  • PARI
    seq(n) = dirmul(vector(n,k,k^2), vector(n,k,moebius(k)));
seq(48)  \\ Gheorghe Coserea, May 11 2016
  • PARI
    jordan(n,k)=my(a=n^k);fordiv(n,i,if(isprime(i),a*=(1-1/(i^k))));a  \\ Roderick MacPhee, May 05 2017
  • Python
    from math import prod
from sympy import factorint
def A007434(n): return prod(p**(e-1<<1)*(p**2-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

Formula

Moebius transform of squares.
Multiplicative with a(p^e) = p^(2e) - p^(2e-2). - Vladeta Jovovic, Jul 26 2001
a(n) = Sum_{d|n} d^2 * mu(n/d). - Benoit Cloitre, Apr 05 2002
a(n) = n^2 * Product_{p|n} (1-1/p^2). - Tom Edgar, Jan 07 2015
a(n) = Sum_{d|n} phi(d)*phi(n/d)*n/d; Sum_{d|n} a(d) = n^2. - Reinhard Zumkeller, Aug 20 2005
Dirichlet generating function: zeta(s-2)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Dirichlet inverse of A046970. - Michael Somos, Jan 11 2014
a(n) = a(n^2)/n^2. - Enrique Pérez Herrero, Sep 14 2010
a(n) = A000010(n) * A001615(n).
If n > 1, then 1 > a(n)/n^2 > 1/zeta(2). - Enrique Pérez Herrero, Jul 14 2011
a(n) = Sum_{d|n} phi(n^2/d)*mu(d)^2. - Enrique Pérez Herrero, Jul 24 2012
a(n) = Sum_{k = 1..n} gcd(k, n)^2 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
a(1) + a(2) + ... + a(n) ~ 1/(3*zeta(3))*n^3 + O(n^2). Lambert series Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x*(1 + x)/(1 - x)^3. - Peter Bala, Dec 23 2013
n * a(n) = A000056(n). - Michael Somos, Mar 20 2004
a(n) = 24 * A115000(n) unless n < 5. - Michael Somos, Aug 12 2008
a(n) = A001065(n) - A134675(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
a(n) = Sum_{k=1..n} gcd(n, k) * phi(gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 15 2018
G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^2 - 1)^2) = 1.81078147612156295224312590448625180897250361794500723589001447178002894356... - Vaclav Kotesovec, Sep 19 2020
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^2 = 1/zeta(3) (A088453). - Amiram Eldar, Oct 12 2020
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*mu(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} gcd(n,k)^2*mu(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} n*phi(gcd(n,k))/gcd(n,k).
a(n) = Sum_{k=1..n} phi(n*gcd(n,k))*mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} phi(n^2/gcd(n,k))*mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = Sum_{k = 1..n} phi(gcd(n, k)^2) = Sum_{d divides n} phi(d^2)*phi(n/d). - Peter Bala, Jan 17 2024
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i)*phi(j). See Tóth, p. 14. - Peter Bala, Jan 29 2024
Conjecture: a(n) = lim_{k->oo} (n^(2*(k + 1)))/A001157(n^k). - Velin Yanev, Dec 04 2024

Extensions

Thanks to Michael Somos for catching an error in this sequence.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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