0,2

Maple procedures Phi_pos_terms and Phi_neg_terms are modeled after the formula given in Lam and Leung paper and they compute correct results for all integers x > 1 and for all n with at most two distinct odd prime factors (that is, up to n=104). Other procedures as in A063698 and A063694.

D. M. Bloom, On the Coefficients of the Cyclotomic Polynomials, Amer.Math.Monthly 75, 372-377, 1968.

T. Y. Lam and K. H. Leung, On the Cyclotomic Polynomial Phi_pq(X), Amer.Math.Monthly 103, 562-564, August-September 1996.

H. W. Lenstra, Vanishing sums of roots of unity, in Proc. Bicentennial Congress Wiskundig Genootschap (Vrije Univ. Amsterdam, 1978), Part II, pp. 249-268.

with(numtheory); [seq(Phi_pos_terms(j, 2), j=0..104)];

inv_p_mod_q := (p, q) -> op(2, op(1, msolve(p*x=1, q))); # Find's p's inverse modulo q.

dilate := proc(nn, x, e) local n, i, s; n := nn; i := 0; s := 0; while(n > 0) do s := s + (((x^e)^i)*(n mod x)); n := floor(n/x); i := i+1; od; RETURN(s); end;

Phi_pos_terms := proc(n, x) local a, m, p, q, e, f, r, s; if(n < 2) then RETURN(x); fi; a := op(2, ifactors(n)); m := nops(a); p := a[1][1]; e := a[1][2]; if(1 = m) then RETURN(((x^(p^e))-1)/((x^(p^(e-1)))-1)); fi; if(2 = m) then q := a[2][1]; f := a[2][2]; r := inv_p_mod_q(p, q)-1; s := inv_p_mod_q(q, p)-1; RETURN( (`if`(0=s, 1, (((x^((s+1)*((q^f)*(p^(e-1)))))-1)/((x^((q^f)*(p^(e-1))))-1)))) * (`if`(0=r, 1, (((x^((r+1)*((p^e)*(q^(f-1)))))-1)/((x^((p^e)*(q^(f-1))))-1)))) ); fi; if((3 = m) and (2 = p)) then if(1 = e) then RETURN(every_other_pos(Phi_pos_terms(n/2, x), x, 0)+every_other_pos(Phi_neg_terms(n/2, x), x, 1)); else RETURN(dilate(Phi_pos_terms((n/(2^(e-1))), x), x, 2^(e-1))); fi; else printf(`Cannot handle argument %a with three or more distinct odd prime factors!\n`, n); RETURN(0); fi; end;

a[n_] := 2^(Flatten[Position[CoefficientList[Cyclotomic[n, x], x], _?Positive]] - 1) // Total; a[0] = 0; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2016 *)

(PARI) a(n)=local(p); if(n<1, 0, p=polcyclo(n); sum(i=0, n, 2^i*(polcoeff(p, i)>0)))

Cf. A013594, A063697 (binary version), A063698 (negative terms), A063670 (nonzero terms).

A019320(n) = a(n) - A063698(n) for up to n=104.

Sequence in context: A153488 A275115 A085399 * A258126 A332211 A353075

Adjacent sequences: A063693 A063694 A063695 * A063697 A063698 A063699

Antti Karttunen, Aug 03 2001

approved