A370252 a(n) is the smallest positive integer k for which there is an identity of the form k*x = Sum_{i=1..m} k_i*g_i(x)^n where k_1, ..., k_m are in Z and g_1(x), ..., g_m(x) are in Z[x].
1, 2, 6, 24, 10, 360, 14, 336, 54, 300, 11, 7920, 39, 2548, 450, 672, 34, 18360, 19, 11400, 882, 484, 23, 2550240, 250, 10140, 162, 15288, 29, 52200, 310, 41664, 2178, 1156, 2450, 403920, 37, 53428, 3042, 159600, 41, 9402120, 43, 124872, 1350
Offset: 1
Examples
a(2) = 2. Since 2x = (x+1)^2 - x^2 - 1, we know a(2) <= 2. Also, a(2) cannot be 1 since the x coefficient of the square of an integer polynomial must be even.
Links
- Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..150 from Daniel Zhu)
- T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Bull. Amer. Math. Soc. 81 (1975), 107-110.
- T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Acta Arith. 29 (1976), 227-250.
- Daniel G. Zhu, A correction to a result of Chinburg and Henriksen on powers of integer polynomials, arXiv:2402.10121 [math.NT], 2024.
Programs
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Python
from itertools import count from sympy import nextprime def A370252(n): c, p = n, 2 while p < n: if n%p: for m in count(2): if (p**m-1)//(p-1) > n: break for r in count(1): q = (p**(m*r)-1)//(p**r-1) if q > n: break if not n % q: c *= p break else: continue if q <= n: break else: c *= p if p&1 or n%6 else p**2 p = nextprime(p) return c # Chai Wah Wu, Mar 10 2024
Formula
a(n) = n*A005729(n)
Comments