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%I M2717 #203 Jan 30 2024 11:42:02
%S 1,3,8,12,24,24,48,48,72,72,120,96,168,144,192,192,288,216,360,288,
%T 384,360,528,384,600,504,648,576,840,576,960,768,960,864,1152,864,
%U 1368,1080,1344,1152,1680,1152,1848,1440,1728,1584,2208,1536
%N Jordan function J_2(n) (a generalization of phi(n)).
%C 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
%C 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
%C 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
%C 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
%C From _Jianing Song_, Apr 06 2019: (Start)
%C 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).
%C For n >= 5, a(n) is divisible by 24. (End)
%C The Del Centina article on page 106 mentions a formula by Halphen denoted by phi(n)T(n). - _Michael Somos_, Feb 05 2021
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%D A. Del Centina, Poncelet's porism: a long story of renewed discoveries, I, Hist. Exact Sci. (2016), v. 70, p. 106.
%D L. E. Dickson (1919, repr. 1971). History of the Theory of Numbers I. Chelsea. p. 147.
%D P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.
%D G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Section 6, Problem 64.
%D M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. 206. Springer-Verlag. p. 11.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007434/b007434.txt">Table of n, a(n) for n = 1..1000</a>
%H Sukumar Das Adhikari and A. Sankaranarayanan, <a href="http://dx.doi.org/10.1016/0022-314X(90)90148-K">On an error term related to the Jordan totient function Jk(n)</a>, Journal of Number Theory Volume 34, Issue 2, February 1990, Pages 178-188.
%H Dorin Andrica and Mihai Piticari, <a href="http://www.emis.de/journals/AUA/acta7/Andrica%20.pdf">On Some Extensions Of Jordan's Arithmetic Functions</a>, Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics - ICTAMI 2003, Alba Iulia.
%H Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, <a href="https://arxiv.org/abs/2308.04751">Hurwitz numbers for reflection groups III: Uniform formulas</a>, arXiv:2308.04751 [math.CO], 2023, see p. 32.
%H F. A. Lewis and others, <a href="http://www.jstor.org/stable/2303350">Problem 4002</a>, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
%H H. Li and T. MacHenry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/MacHenry/machenry7.html">Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences</a>, J. Int. Seq. 16 (2013) #13.3.5, example 38.
%H MathOverflow, <a href="http://mathoverflow.net/q/84571">Averages of euler-phi function and similar</a>.
%H Carl-Fredrik Nyberg-Brodda, <a href="https://arxiv.org/abs/2312.11258">On congruence subgroups of SL_2(Z[1/p]) generated by two parabolic elements</a>, arXiv:2312.11258 [math.GR], 2023.
%H Nittiya Pabhapote and Vichian Laohakosol, <a href="http://dx.doi.org/10.1155/2010/648165">Combinatorial Aspects of the Generalized Euler's Totient</a>, International Journal of Mathematics and Mathematical Sciences, Volume 2010 (2010), Article ID 648165, 15 p.
%H Wolfgang Schramm, <a href="http://www.emis.de/journals/INTEGERS/papers/i50/i50.Abstract.html">The Fourier transform of functions of the greatest common divisor</a>, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H S. Thajoddin and S. Vangipuram, <a href="https://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a79_1156.pdf">A Note On Jordan's Totient Function</a>, Indian J. Pure Appl. Math, 1988.
%H László Tóth, <a href="http://arxiv.org/abs/1310.7053">Multiplicative arithmetic functions of several variables: a survey</a>, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F Moebius transform of squares.
%F Multiplicative with a(p^e) = p^(2e) - p^(2e-2). - _Vladeta Jovovic_, Jul 26 2001
%F a(n) = Sum_{d|n} d^2 * mu(n/d). - _Benoit Cloitre_, Apr 05 2002
%F a(n) = n^2 * Product_{p|n} (1-1/p^2). - _Tom Edgar_, Jan 07 2015
%F a(n) = Sum_{d|n} phi(d)*phi(n/d)*n/d; Sum_{d|n} a(d) = n^2. - _Reinhard Zumkeller_, Aug 20 2005
%F Dirichlet generating function: zeta(s-2)/zeta(s). - _Franklin T. Adams-Watters_, Sep 11 2005
%F Dirichlet inverse of A046970. - _Michael Somos_, Jan 11 2014
%F a(n) = a(n^2)/n^2. - _Enrique Pérez Herrero_, Sep 14 2010
%F a(n) = A000010(n) * A001615(n).
%F If n > 1, then 1 > a(n)/n^2 > 1/zeta(2). - _Enrique Pérez Herrero_, Jul 14 2011
%F a(n) = Sum_{d|n} phi(n^2/d)*mu(d)^2). - _Enrique Pérez Herrero_, Jul 24 2012
%F a(n) = Sum_{k = 1..n} gcd(k, n)^2 * cos(2*Pi*k/n). - _Enrique Pérez Herrero_, Jan 18 2013
%F 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
%F n * a(n) = A000056(n). - _Michael Somos_, Mar 20 2004
%F a(n) = 24 * A115000(n) unless n < 5. - _Michael Somos_, Aug 12 2008
%F a(n) = A001065(n) - A134675(n). - Conjectured by _John Mason_ and proved by _Max Alekseyev_, Jan 07 2015
%F 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
%F G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^k)/(1 - x^k)^3. - _Ilya Gutkovskiy_, Oct 24 2018
%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^2 - 1)^2) = 1.81078147612156295224312590448625180897250361794500723589001447178002894356... - _Vaclav Kotesovec_, Sep 19 2020
%F Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^2 = 1/zeta(3) (A088453). - _Amiram Eldar_, Oct 12 2020
%F From _Richard L. Ollerton_, May 09 2021: (Start)
%F a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*mu(gcd(n,k))/phi(n/gcd(n,k)).
%F a(n) = Sum_{k=1..n} gcd(n,k)^2*mu(n/gcd(n,k))/phi(n/gcd(n,k)).
%F a(n) = Sum_{k=1..n} n*phi(gcd(n,k))/gcd(n,k).
%F a(n) = Sum_{k=1..n} phi(n*gcd(n,k))*mu(n/gcd(n,k))^2.
%F 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)
%F 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
%F a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i)*phi(j). See Tóth, p. 14. - _Peter Bala_, Jan 29 2024
%e 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.
%e 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 + ...
%p 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)
%p A007434 := proc(n)
%p add(d^2*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
%p end proc: # _R. J. Mathar_, Nov 03 2015
%t 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 *)
%t a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ n/d], {d, Divisors @ n}]]; (* _Michael Somos_, Jan 11 2014 *)
%t a[ n_] := If[ n < 2, Boole[ n == 1], n^2 (Times @@ ((1 - 1/#[[1]]^2) & /@ FactorInteger @ n))]; (* _Michael Somos_, Jan 11 2014 *)
%t jordanTotient[n_Integer?Positive, r_:1] := DirichletConvolve[MoebiusMu[K], K^r, K, n]; Table[jordanTotient[n, 2], {n, 48}] (* _Jan Mangaldan_, Jun 03 2016 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * moebius(n / d)))}; /* _Michael Somos_, Mar 20 2004 */
%o (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X) / (1 - X*p^2))[n])}; /* _Michael Somos_, Jan 11 2014 */
%o (PARI) seq(n) = dirmul(vector(n,k,k^2), vector(n,k,moebius(k)));
%o seq(48) \\ _Gheorghe Coserea_, May 11 2016
%o (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
%o (Haskell)
%o a007434 n = sum $ zipWith3 (\x y z -> x * y * z)
%o tdivs (reverse tdivs) (reverse divs)
%o where divs = a027750_row n; tdivs = map a000010 divs
%o -- _Reinhard Zumkeller_, Nov 24 2012
%o (Python)
%o from math import prod
%o from sympy import factorint
%o 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
%Y Cf. A000056, A000290, A001065, A001615, A027750, A046970, A115000, A134675.
%Y Cf. A059379 and A059380 (triangle of values of J_k(n)).
%Y Cf. A000010 (J_1), this sequence (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).
%Y Cf. A002117, A088453, A301875, A301876, A321879 (partial sums).
%K nonn,easy,nice,mult,look
%O 1,2
%A _N. J. A. Sloane_
%E Thanks to _Michael Somos_ for catching an error in this sequence.
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