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a(n) = J_2(n) / 24.
7

%I #30 Sep 08 2022 08:45:23

%S 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,

%T 32,40,36,48,36,57,45,56,48,70,48,77,60,72,66,92,64,98,75,96,84,117,

%U 81,120,96,120,105,145,96,155,120,144,128,168,120,187,144,176,144,210,144

%N a(n) = J_2(n) / 24.

%C 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.

%C 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.)

%C 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.

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

%C 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

%H G. C. Greubel, <a href="/A115000/b115000.txt">Table of n, a(n) for n = 5..5000</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F A007434(n) = 24 * a(n) unless n<5. - _Michael Somos_, Aug 12 2008

%e 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 + ...

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

%o (PARI) {a(n) = if( n<5, 0, sumdiv(n, d, d^2 * moebius(n / d)) / 24)}; /* _Michael Somos_, Aug 12 2008 */

%o (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 */

%Y Cf. A007434.

%K nonn

%O 5,3

%A _Franklin T. Adams-Watters_, Dec 10 2005

%E More terms from _Michael Somos_, Aug 12 2008