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%I #25 Jun 13 2021 07:17:35
%S 2,0,0,1,0,2,0,1,1,2,0,2,0,2,2,1,0,2,0,2,2,2,0,2,1,2,1,2,0,2,0,1,2,2,
%T 2,2,0,2,2,2,0,2,0,2,2,2,0,2,1,2,2,2,0,2,2,2,2,2,0,2,0,2,2,1,2,2,0,2,
%U 2,2,0,2,0,2,2,2,2,2,0,2,1,2,0,2,2,2,2,2,0,2,2,2,2,2,2,2,0,2,2,2,0,2,0,2,3
%N Consider the coefficients in the expansion of the n-th cyclotomic polynomial. a(n) is the difference between the extremes.
%C Conjecture: every number occurs in this sequence. This is based on the fact that every integer is a coefficient in the expansion of a cyclotomic polynomial. See Chun-Gang Ji and Wei-Ping Li link below.
%C First occurrence of k, for k>=0: 2, 4, 1, 105, 330, 385, 770, 1365, 1995, 1785, 3570, 5610, 2805, 6279, 3135, 14245, ..., see A345080.
%H Robert G. Wilson v, <a href="/A345079/b345079.txt">Table of n, a(n) for n = 1..10000</a>
%H Chun-Gang Ji and Wei-Ping Li, <a href="https://doi.org/10.1016/j.disc.2007.10.009">Values of coefficients of cyclotomic polynomials</a>, Discrete Mathematics, Vol. 308, No. 23 (2008), 5860-5863.
%H Emma Lehmer, <a href="http://dx.doi.org/10.1090/S0002-9904-1936-06309-3">On the magnitude of the coefficients of the cyclotomic polynomial</a>, Bull. Amer. Math. Soc. 42 (1936), 389-392.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic Polynomial</a>.
%F a(n) = 0 if and only if n is prime.
%F a(n) = 1 if and only if n = p^e with prime p, e >= 2 (A246547). The "if" part is obvious. For the converse, note that Phi_n(1) = 1 if and only if n is not a prime power (A246655). If n is not a prime power and Phi_n has only nonnegative coefficients, then Phi_n(1) = 1 implies that Phi_n is a monomial, which is impossible.
%e a(1) = 2. The expansion of the 1st cyclotomic polynomial, Phi_1(x) = x - 1; the difference between 1 and -1 is 2;
%e a(2) = 0. The expansion of the 2nd cyclotomic polynomial, Phi_2(x) = x + 1; the difference between 1 and 1 is 0;
%e a(105) = 3. The expansion of the 105th cyclotomic polynomial, Phi_105(x) = x^48 + x^47 + ... - x^8 - 2x^7 - x^6 + ... + 1; the difference between 1 and -2 is 3; etc.
%t a[n_] := Block[{b = Union[ CoefficientList[ Cyclotomic[n, x], x]]}, b[[-1]] - b[[1]]]; Array[a, 105]
%o (PARI) A345079(n) = my(v=Vec(polcyclo(n))); vecmax(v) - vecmin(v)
%Y Cf. A013595, A345080, A230798, A246547, A246655.
%K nonn
%O 1,1
%A _Jianing Song_ and _Robert G. Wilson v_, Jun 04 2021
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