Search: id:A007434
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%I A007434 M2717
%S A007434 1,3,8,12,24,24,48,48,72,72,120,96,168,144,192,192,288,216,360,
%T A007434 288,384,360,528,384,600,504,648,576,840,576,960,768,960,864,1152,
%U A007434 864,1368,1080,1344,1152,1680,1152,1848,1440,1728,1584,2208,1536
%N A007434 Jordan function J_2(n) (a generalization of phi(n)).
%C A007434 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 A007434 A000056(n)=n*a(n). - Michael Somos Mar 20 2004
%C A007434 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 iff GCD(u,v,n)=1. 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2005
%C A007434 The weight of the n-th polynomial for the analog of cyclotomic polynomials 
               for elliptic divisibility sequences. That is, let 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
%D A007434 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007434 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%D A007434 F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 
               9, Nov. 1942, pp. 618-619.
%D A007434 G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 
               1924, reprinted 1972), Part Eight, Chap. 1, Section 6, Problem 64.
%H A007434 T. D. Noe, <a href="b007434.txt">Table of n, a(n) for n=1..1000</a>
%H A007434 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A007434 Moebius transform of squares.
%F A007434 Multiplicative with a(p^e) = p^(2e)-p^(2e-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Jul 26 2001
%F A007434 a(n)=sum(d|n, d^2*mu(n/d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 05 2002
%F A007434 a(n) = Sum(phi(d)*phi(n/d)*n/d: d divides n); Sum(a(d): d divides n) 
               = n^2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 
               20 2005
%F A007434 Dirichlet generating function: zeta(s-2)/zeta(s). - Franklin T. Adams-Watters, 
               Sep 11 2005.
%p A007434 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)
%o A007434 (PARI) a(n)=if(n<1,0,sumdiv(n,d,d^2*moebius(n/d)))
%Y A007434 Cf. A000290. Cf. A059379 and A059380 (triangle of values of J_k(n)), 
               A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
%Y A007434 A115000(n) = a(n) / 24 unless n<5. - Michael Somos Aug 12 2008
%Y A007434 Sequence in context: A014255 A022407 A158022 this_sequence A128303 A123906 
               A065970
%Y A007434 Adjacent sequences: A007431 A007432 A007433 this_sequence A007435 A007436 
               A007437
%K A007434 nonn,easy,nice,mult
%O A007434 1,2
%A A007434 N. J. A. Sloane (njas(AT)research.att.com).
%E A007434 Thanks to Michael Somos for catching an error in this sequence.

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