A063696 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

0, 2, 3, 7, 5, 31, 5, 127, 17, 73, 21, 2047, 17, 8191, 85, 297, 257, 131071, 65, 524287, 273, 4681, 1365, 8388607, 257, 1082401, 5461, 262657, 4369, 536870911, 387, 2147483647, 65537, 1198665, 87381, 17454241, 4097, 137438953471, 349525

Offset: 0

Author

Antti Karttunen, Aug 03 2001

Keywords

• nonn

Comments

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.

Links

• 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.
• Index entries for cyclotomic polynomials, positions of coefficients

Crossrefs

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

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

Programs

Maple

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;

Mathematica

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)))