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

1,2

Comments

a(n) is a multiple of n and a divisor of n!.

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.

Formula

a(n) = n*A005729(n)

Example

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.

Prog

(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

Crossrefs

Cf. A005729, A005730, A005731.

Sequence in context: A033642 A324528 A334795 * A181952 A345902 A256270

Adjacent sequences: A370249 A370250 A370251 * A370253 A370254 A370255

Keyword

nonn,easy

Author

Daniel Zhu, Feb 17 2024

Status

approved