A063696 -id:A063696 - cp's OEIS frontend

A063697 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), A063696 in binary.

0, 10, 11, 111, 101, 11111, 101, 1111111, 10001, 1001001, 10101, 11111111111, 10001, 1111111111111, 1010101, 100101001, 100000001, 11111111111111111, 1000001, 1111111111111111111, 100010001, 1001001001001, 10101010101

Offset: 0

Antti Karttunen, Aug 03 2001

			Phi_15(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1, thus the 1-bits of a(15) are at positions 0,3,5 and 8, thus we get a(15) = 100101001

Index entries for cyclotomic polynomials, positions of coefficients

Cf. A063699, A063671, A063672.

Maple
```
map(convert, A063696,binary);
```

a(0) corrected by Sean A. Irvine, May 08 2023

A019320 Cyclotomic polynomials at x=2.

Original entry on oeis.org

2, 1, 3, 7, 5, 31, 3, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 57, 524287, 205, 2359, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681

Offset: 0

Simon Plouffe

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook)
Index entries for cyclotomic polynomials, values at X

a(n) = A063696(n) - A063698(n) for up to n=104.
Same sequence in binary: A063672.
Cf. A054432, A020501, A034268.

with(numtheory,cyclotomic); f := n->subs(x=2,cyclotomic(n,x)); seq(f(i),i=0..64);

Join[{2}, Table[Cyclotomic[n, 2], {n, 1, 40}]] (* Jean-François Alcover, Jun 14 2013 *)

vector(20,n,polcyclo(n,2)) \\ Charles R Greathouse IV, May 18 2011

(lcm_{k=1..n} (2^k - 1))/lcm_{k=1..n-1} (2^k - 1), n > 1. - Vladeta Jovovic, Jan 20 2002
Let b(1) = 1 and b(n+1) = lcm(b(n), 2^n-1) then Phi(n,2) = b(n+1)/b(n) = a(n). - Thomas Ordowski, May 08 2013
a(0) = 2; for n > 0, a(n) = (2^n-1)/gcd(a(0)*a(1)*...*a(n-1), 2^n-1). - Thomas Ordowski, May 11 2013

A013594 Smallest order of cyclotomic polynomial containing n or -n as a coefficient.

Original entry on oeis.org

0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 11305, 17255, 17255, 20615, 20615, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565

Offset: 1

N. J. A. Sloane

This sequence is infinite - see the Lang reference.

An alternative version would start with 1 rather than 0.

			a(2)=105 because cyclotomic(105) contains "-2" as coefficient, but for n < 105 cyclotomic(n) does not contain 2 or -2.
x^105 - 1 = ( - 1 + x)(1 + x + x^2)(1 + x + x^2 + x^3 + x^4)(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)(1 - x + x^3 - x^4 + x^5 - x^7 + x^8)(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12)(1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + x^17 - x^18 + x^19 - x^23 + x^24)(1 + x + x^2 - x^5 - x^6 - 2x^7 - x^8 - x^9 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 - x^20 - x^22 - x^24 - x^26 - x^28 + x^31 + x^32 + x^33 + x^34 + x^35 + x^36 - x^39 - x^40 - 2x^41 - x^42 - x^43 + x^46 + x^47 + x^48)

Bateman, C. Pomerance and R. C. Vaughan, Colloq. Math. Soc. Janos Bolyai, 34 (1984), 171-202.
S. Lang, Algebra: 3rd edition, Addison-Wesley, 1993, p. 281.
Maier, Prog. Math. 85 (Birkhaueser), 1990, 349-366.
Maier, Prog. Math. 139 (Birkhaueser) 1996, 633-638.

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Arnold and M. Monagan, Calculating cyclotomic polynomials of very large height.
P. Erdős and R. C. Vaughan, Bounds for the r-th coefficients of cyclotomic polynomials, J. London Math. Soc. (2) 8 (1974), 393-400 (MR50 #9835; Zentralblatt 295.10014).
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Maier, Cyclotomic polynomials with large coefficients, Acta Arith. 64 (1993), 227-235.
H. Maier, Cyclotomic polynomials whose orders contain many prime factors, Period. Math. Hungar. 43 (2001), 155-169.
H. L. Montgomery and R. C. Vaughan, The order of magnitude of the mth coefficients of cyclotomic polynomials, Glasgow Math. J. 27 (1985), 143-159.
R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).
M. Wallner, Lattice Path Combinatorics, Diplomarbeit, Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien, 2013.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A046887, A013595, A013596, A063696, A063698, A134518, A137979.

Table[Position[Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 10000}], j][[1]], {j, 1, 10}] (* Ian Miller, Feb 25 2008 *)

nm=6545; m=0; forstep(n=1, nm, 2, if(issquarefree(n), p=polcyclo(n); o=poldegree(p); for(k=0, o, a=abs(polcoeff(p, k)); if(a>m, m=a; print([m, n, factor(n)])))))

More terms from Eric W. Weisstein

Further terms from T. D. Noe, Oct 29 2007

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).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

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.

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

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

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;

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 *)

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

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

Original entry on oeis.org

2, 3, 3, 7, 5, 31, 7, 127, 17, 73, 31, 2047, 21, 8191, 127, 443, 257, 131071, 73, 524287, 341, 7003, 2047, 8388607, 273, 1082401, 8191, 262657, 5461, 536870911, 443, 2147483647, 65537, 1797851, 131071, 26181091, 4161, 137438953471, 524287

Offset: 0

Antti Karttunen, Aug 03 2001

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 1A005420(n)=2^n-1 (i.e., 2^n-1 is prime). - M. F. Hasler, Apr 30 2007

a(0) could also be 1. - T. D. Noe, Oct 29 2007

Cf. A013594.
a(n) = A063696(n) (the positive terms) + A063698(n) (the negative terms).
This sequence in binary: A063671.
Cf. A005420.

[seq(Phi_pos_terms(j,2)+Phi_neg_terms(j,2),j=0..104)];

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 *)

A063670(n)=local(p=polcyclo(n+!n)); if(n,sum(i=0, n, (polcoeff(p, i)<>0)<M. F. Hasler, Apr 30 2007

a(n) = subst(apply(x->x!=0, polcyclo(n,'x)), 'x, 2);  \\ Gheorghe Coserea, Nov 04 2016

cp's OEIS Frontend

A063697 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), A063696 in binary.

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A019320 Cyclotomic polynomials at x=2.

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A013594 Smallest order of cyclotomic polynomial containing n or -n as a coefficient.

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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).

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Maple

Mathematica

PARI

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

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Keywords

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Maple

Mathematica

PARI

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