5,3

The Jordan function J_m(n) can be defined as multiplicative with J_m(p^e) = (p^m-1)*p^(m*(e-1)). Cf. A059379.

Looking at the sequences J_m(n) for fixed m, one is struck by the fact that all but a few early terms have a common factor, given in A079612. I will refer to this sequence as K(n), following the notation in the paper by Vaughan and Wooley. (The alternate lambda^*(n) in the comment for A006863 is too awkard.)

In fact, K(m) not only divides J_m(n) for all but finitely many n; it also divides Sum_{k=1}^n J_m(k) for all but finitely many n.

J_1(n) = phi(n) and phi(n)/2 and Sum_{k=1}^n phi(n)/2 are A023022 and A046657.

The weight of the nth reduced polynomial for the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, 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. Let c2 = b2^4*b4, c3 = b3^3, c4 = b4^2 and cn = bn for n>4. Let weight of c2 = c3 = c4 = 1. Then c5 = c2 - c3, c6 = c5 - c4, c7 = c6*c3 - c5*c4 and so on and the weight of cn is a(n) for n>4. - Michael Somos Aug 12 2008

(PARI) {a(n) = if(n<5, 0, sumdiv(n, d, d^2 * moebius(n / d)) / 24)} - Michael Somos Aug 12 2008

A007434(n) = 24 * a(n) unless n<5. - Michael Somos Aug 12 2008

Sequence in context: A114092 A082500 A059292 this_sequence A036825 A035574 A036819

Adjacent sequences: A114997 A114998 A114999 this_sequence A115001 A115002 A115003

nonn

Franklin T. Adams-Watters, Dec 10 2005

More terms from Michael Somos Aug 12 2008