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 nth 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

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.yu), 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).

Adjacent sequences: A007431 A007432 A007433 this_sequence A007435 A007436 A007437

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

Sequence in context: A103888 A014255 A022407 this_sequence A128303 A123906 A065970

nonn,easy,nice,mult

njas

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