A063670 -id:A063670 - cp's OEIS frontend

A063671 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), A063670 in binary.

10, 11, 11, 111, 101, 11111, 111, 1111111, 10001, 1001001, 11111, 11111111111, 10101, 1111111111111, 1111111, 110111011, 100000001, 11111111111111111, 1001001, 1111111111111111111, 101010101, 1101101011011, 11111111111

Offset: 0

Antti Karttunen, Aug 03 2001

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

			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 0,1,3,4,5,7 and 8, thus we get a(15) = 110111011.

Cf. A063672, A063697, A063699.

Cf. A013595.

Maple
```
map(convert, A063670,binary);
```

a[n_] := FromDigits[Abs[CoefficientList[Cyclotomic[n, x], x]]]; a[0]=10;Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Nov 02 2016 *)

a(n) = A063697(n) (the positive terms) + A063699(n) (the negative terms) (computed in any base, up to n=104).

A128997 indices k such that A063670(k) differs from A005420(k).

Original entry on oeis.org

11, 12, 15, 20, 21, 22, 23, 24, 25, 28, 29, 30, 33, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

Offset: 1

M. F. Hasler, Apr 30 2007

nonn

Primes in this sequence are those for which A005420(n) < 2^n-1, i.e. 2^n-1 is not prime.

Cf. A005420, A063670.

t=1;vector(100,i,until( A063670(t++) != A005420(t),);t)

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

Original entry on oeis.org

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

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 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.
Index entries for cyclotomic polynomials, positions of coefficients

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

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

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 *)

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

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A063671 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), A063670 in binary.

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A128997 indices k such that A063670(k) differs from A005420(k).

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

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