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A063670 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

%I A063670 #29 Nov 05 2016 08:00:58
%S A063670 2,3,3,7,5,31,7,127,17,73,31,2047,21,8191,127,443,257,131071,73,
%T A063670 524287,341,7003,2047,8388607,273,1082401,8191,262657,5461,536870911,
%U A063670 443,2147483647,65537,1797851,131071,26181091,4161,137438953471,524287
%N A063670 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.
%C A063670 a(n) = 2^n-1 whenever n is prime. It seems as if a(n) >= A005420(n) for all n (checked up to 200), with equality for all 1<n<20 except {11,12,15} and whenever A005420(n)=2^n-1 (i.e., 2^n-1 is prime). - _M. F. Hasler_, Apr 30 2007
%C A063670 a(0) could also be 1. - _T. D. Noe_, Oct 29 2007
%H A063670 M. F. Hasler, <a href="/A063670/b063670.txt">Table of n, a(n) for n = 0..1000</a>
%H A063670 <a href="/index/Cy#CyclotomicPolynomialsCoefficients">Index entries for cyclotomic polynomials, positions of coefficients</a>
%p A063670 [seq(Phi_pos_terms(j,2)+Phi_neg_terms(j,2),j=0..104)];
%t A063670 a[n_] := FromDigits[ If[# != 0, 1, 0]& /@ CoefficientList[ Cyclotomic[n, x], x], 2]; a[0] = 2; Table[a[n], {n, 0, 38}] (* _Jean-François Alcover_, Dec 11 2012 *)
%o A063670 (PARI) A063670(n)=local(p=polcyclo(n+!n)); if(n,sum(i=0, n, (polcoeff(p, i)<>0)<<i),2) \\ _M. F. Hasler_, Apr 30 2007
%o A063670 (PARI) a(n) = subst(apply(x->x!=0, polcyclo(n,'x)), 'x, 2);  \\ _Gheorghe Coserea_, Nov 04 2016
%Y A063670 Cf. A013594.
%Y A063670 a(n) = A063696(n) (the positive terms) + A063698(n) (the negative terms).
%Y A063670 This sequence in binary: A063671.
%Y A063670 Cf. A005420.
%K A063670 nonn,base
%O A063670 0,1
%A A063670 _Antti Karttunen_, Aug 03 2001