1,2

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

A000056(n)=n*a(n). - Michael Somos Mar 20 2004

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

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.

G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Section 6, Problem 64.

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

N. J. A. Sloane, Transforms

Moebius transform of squares.

Multiplicative with a(p^e) = p^(2e)-p^(2e-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 26 2001

a(n)=sum(d|n, d^2*mu(n/d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002

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

Dirichlet generating function: zeta(s-2)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005.

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)

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, d^2*moebius(n/d)))

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).

A115000(n) = a(n) / 24 unless n<5. - Michael Somos Aug 12 2008

Sequence in context: A014255 A022407 A158022 this_sequence A128303 A123906 A065970

Adjacent sequences: A007431 A007432 A007433 this_sequence A007435 A007436 A007437

nonn,easy,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).

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