A063672 -id:A063672 - cp's OEIS frontend

A019320 Cyclotomic polynomials at x=2.

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

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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).

A063699 Positions of negative coefficients in cyclotomic polynomial Phi_n(x), A063698 in binary.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 10, 0, 0, 0, 1010, 0, 100, 0, 101010, 10010010, 0, 0, 1000, 0, 1000100, 100100010010, 1010101010, 0, 10000, 0, 101010101010, 0, 10001000100, 0, 111000, 0, 0, 10010010010010010010, 1010101010101010, 100001010010100101000010

Offset: 0

Antti Karttunen, Aug 03 2001

			E.g. 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 1,4 and 7, thus we get a(15) = 10010010

Index entries for cyclotomic polynomials, positions of coefficients

Cf. A063697, A063671, A063672.

Maple
```
map(convert, A063698,binary);
```

A252502 Number of digits of Phi_n(10), or number of digits in base b of Phi_n(b), where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 2, 3, 3, 5, 2, 7, 5, 7, 4, 11, 4, 13, 6, 8, 9, 17, 6, 19, 8, 12, 10, 23, 8, 21, 12, 19, 12, 29, 9, 31, 17, 20, 16, 24, 12, 37, 18, 24, 16, 41, 13, 43, 20, 24, 22, 47, 16, 43, 20, 32, 24, 53, 18, 40, 24, 36, 28, 59, 17, 61, 30, 36, 33, 48, 21, 67, 32, 44, 25, 71, 24

Offset: 1

Eric Chen, Dec 17 2014

nonn

a(n) = phi(n) if and only if the number of distinct prime factors of n (A001221(n)) is even, a(n) = phi(n) + 1 if and only if the number of distinct prime factors of n (A001221(n)) is odd, where phi is Euler's totient function.

			Values of phi_n(b) written in base b are #, 11, 111, 101, 11111, #1, 1111111, 10001, 1001001, #0#1, 11111111111, ##01, ..., where # represents b - 1.

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A000010, A019328, A063672, A206225.

a252502[n_] := Array[Total@DigitCount[Cyclotomic[#, 10]] &, n]; a252502[72] (* Michael De Vlieger, Dec 21 2014 *)

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A019320 Cyclotomic polynomials at x=2.

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

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

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A063699 Positions of negative coefficients in cyclotomic polynomial Phi_n(x), A063698 in binary.

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A252502 Number of digits of Phi_n(10), or number of digits in base b of Phi_n(b), where Phi is the cyclotomic polynomial.

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