A005729 a(n) is the smallest positive integer a for which there is an identity of the form a*n*x = Sum_{i=1..m} ai*gi(x)^n where a1, ..., am are in Z and g1(x), ..., gm(x) are in Z[x]. (Formerly M1557)

1, 1, 2, 6, 2, 60, 2, 42, 6, 30, 1, 660, 3, 182, 30, 42, 2, 1020, 1, 570, 42, 22, 1, 106260, 10, 390, 6, 546, 1, 1740, 10, 1302, 66, 34, 70, 11220, 1, 1406, 78, 3990, 1, 223860, 1, 2838, 30, 46, 1, 4994220, 14, 210, 102, 390, 1, 54060, 110, 546, 798, 58, 1, 21455940

Offset

1,3

Comments

Previous name was: From polynomial identities.

The originally published terms of this sequence were incorrect for a small number of n, the smallest of which is n=14 (see the paper of Zhu for more details). - Daniel Zhu, Feb 16 2024

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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 Arithmetica, 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) = A005730(n)*A005731(n).

Prog

(PARI) expa(p, n) = {if (p % 2, return (1)); if (n % 6, return (1)); 2; }
expb(p, n) = {expo = 0; r = 1; ok = 1; while (ok, m = 2; while ((ps = (p^(m*r)-1)/(p^r-1)) <= n, if (n % ps == 0, expo = 1; break); m++; ); if (m==2, ok = 0); if (expo, break); r++; ); expo; }
expp(p, n) = if (n % p, expb(p, n), expa(p, n));
a(n) = {my(vp = primes(primepi(n-1))); prod(k=1, #vp, vp[k]^expp(vp[k], n)); } \\ Michel Marcus, Apr 27 2016 [Corrected by Daniel Zhu, Feb 16 2024]
(Python)
from itertools import count
from sympy import nextprime
def A005729(n):
    c, p = 1, 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. A005730, A005731.

Sequence in context: A284577 A122018 A363395 * A271504 A086660 A271503

Adjacent sequences: A005726 A005727 A005728 * A005730 A005731 A005732

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Jan 24 2005

New name from Michel Marcus, Apr 27 2016

Errors in name and terms a(14), a(28), and a(56) corrected by Daniel Zhu, Feb 16 2024

Status

approved