A115000 a(n) = J_2(n) / 24.

1, 1, 2, 2, 3, 3, 5, 4, 7, 6, 8, 8, 12, 9, 15, 12, 16, 15, 22, 16, 25, 21, 27, 24, 35, 24, 40, 32, 40, 36, 48, 36, 57, 45, 56, 48, 70, 48, 77, 60, 72, 66, 92, 64, 98, 75, 96, 84, 117, 81, 120, 96, 120, 105, 145, 96, 155, 120, 144, 128, 168, 120, 187, 144, 176, 144, 210, 144

Offset: 5

Author

Franklin T. Adams-Watters, Dec 10 2005

Keywords

• nonn

Comments

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 awkward.)

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 n-th elliptic division polynomial -- 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. Then c5 = c2 - c3, c6 = c5 - c4, c7 = c6*c3 - c5*c4 and so on. Let the weight of c2, c3, c4 each be 1 and weight of a product is sum of the weights of the factors. The weight of cn is a(n) for n>4. - Michael Somos, Aug 12 2008

Examples

			G.f.: x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 5*x^11 + 4*x^12 + 7*x^13 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 5..5000
• Index to divisibility sequences

Crossrefs

Cf. A007434.

Programs

Magma

function a(n) return n lt 5 select 0 else Dimension( ModularForms( Gamma1(n), 2)) - Dimension( ModularForms( Gamma1(n), 1)); end function; /* Michael Somos, Aug 05 2014 */

Mathematica

a[n_] := DivisorSum[n, #^2*MoebiusMu[n/#]&]/24; Table[a[n], {n, 5, 80}] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

PARI

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

Formula

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

Extensions

More terms from Michael Somos, Aug 12 2008