A063698 - OEIS

A063698

Positions of negative coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal. (The constant term in the least significant bit (bit-0), the term x in the next bit (bit-1) and so on).

0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 10, 0, 4, 0, 42, 146, 0, 0, 8, 0, 68, 2322, 682, 0, 16, 0, 2730, 0, 1092, 0, 56, 0, 0, 599186, 43690, 8726850, 64, 0, 174762, 9585810, 4112, 0, 792, 0, 279620, 2101256, 2796202, 0, 256, 0, 32800, 2454267026, 4473924, 0, 512

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OFFSET

0,7

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 A063696 and A063694.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..3003

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. 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

MAPLE

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

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

MATHEMATICA

a[n_] := 2^(Flatten[Position[CoefficientList[Cyclotomic[n, x], x], _?Negative]] - 1) // Total; a[0] = 0; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 05 2016 *)

PROG

(PARI) a(n)=my(p); if(n<1, 0, p=polcyclo(n); sum(i=0, n, 2^i*(polcoeff(p, i)<0))) \\ Michel Marcus, Mar 05 2016

CROSSREFS

A013594, A063696 gives the positions of the positive and A063697 the nonzero terms. This sequence in binary: A063699. A019320[n] = A063696[n]-A063698[n] for up to n=104

Sequence in context: A365437 A181502 A223154 * A348030 A287466 A288014

Adjacent sequences: A063695 A063696 A063697 * A063699 A063700 A063701

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 03 2001

STATUS

approved