A086374 Number of factors over Q in the factorization of T_n(x) + 1 where T_n(x) is the Chebyshev polynomial of the first kind.
1, 2, 3, 2, 3, 4, 3, 2, 5, 4, 3, 4, 3, 4, 7, 2, 3, 6, 3, 4, 7, 4, 3, 4, 5, 4, 7, 4, 3, 8, 3, 2, 7, 4, 7, 6, 3, 4, 7, 4, 3, 8, 3, 4, 11, 4, 3, 4, 5, 6, 7, 4, 3, 8, 7, 4, 7, 4, 3, 8, 3, 4, 11, 2, 7, 8, 3, 4, 7, 8, 3, 6, 3, 4, 11, 4, 7, 8, 3, 4, 9, 4, 3, 8, 7, 4, 7, 4, 3, 12, 7, 4, 7, 4, 7, 4, 3, 6, 11, 6, 3, 8, 3
Offset: 1
Examples
a(6) = 4 because T_6(x)+1 = 32x^6-48x^4+18x^2 = x^2*(4x^2-3)^2.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..2049
Programs
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PARI
p2 = 1; p1 = x; for (n = 1, 103, p = 2*x*p1 - p2; f = factor(p1 + 1); print(sum(i = 1, matsize(f)[1], f[i, 2]), " "); p2 = p1; p1 = p); \\ David Wasserman, Mar 03 2005
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PARI
A086374(n) = {vecsum(factor(polchebyshev(n, 1, x)+1)[, 2])}; \\ Antti Karttunen, Sep 27 2018, after Andrew Howroyd's program for A086369
Formula
If p is an odd prime then a(p) = 3.
Extensions
More terms from David Wasserman, Mar 03 2005