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%I M1557 #46 Mar 11 2024 01:55:45
%S 1,1,2,6,2,60,2,42,6,30,1,660,3,182,30,42,2,1020,1,570,42,22,1,106260,
%T 10,390,6,546,1,1740,10,1302,66,34,70,11220,1,1406,78,3990,1,223860,1,
%U 2838,30,46,1,4994220,14,210,102,390,1,54060,110,546,798,58,1,21455940
%N 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].
%C Previous name was: From polynomial identities.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Chai Wah Wu, <a href="/A005729/b005729.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..150 from Daniel Zhu)
%H T. Chinburg and M. Henriksen, <a href="http://dx.doi.org/10.1090/S0002-9904-1975-13657-3">Sums of k-th powers in the ring of polynomials with integer coefficients</a>, Bull. Amer. Math. Soc., 81 (1975), 107-110.
%H T. Chinburg and M. Henriksen, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa29/aa2932.pdf">Sums of k-th powers in the ring of polynomials with integer coefficients</a>, Acta Arithmetica, 29 (1976), 227-250.
%H Daniel G. Zhu, <a href="https://arxiv.org/abs/2402.10121">A correction to a result of Chinburg and Henriksen on powers of integer polynomials</a>, arXiv:2402.10121 [math.NT], 2024.
%F a(n) = A005730(n)*A005731(n).
%o (PARI) expa(p, n) = {if (p % 2, return (1)); if (n % 6, return (1)); 2;}
%o 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;}
%o expp(p, n) = if (n % p, expb(p, n), expa(p, n));
%o 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]
%o (Python)
%o from itertools import count
%o from sympy import nextprime
%o def A005729(n):
%o c, p = 1, 2
%o while p < n:
%o if n%p:
%o for m in count(2):
%o if (p**m-1)//(p-1) > n:
%o break
%o for r in count(1):
%o q = (p**(m*r)-1)//(p**r-1)
%o if q > n:
%o break
%o if not n % q:
%o c *= p
%o break
%o else:
%o continue
%o if q <= n:
%o break
%o else:
%o c *= p if p&1 or n%6 else p**2
%o p = nextprime(p)
%o return c # _Chai Wah Wu_, Mar 10 2024
%Y Cf. A005730, A005731.
%K nonn,nice,easy
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Jan 24 2005
%E New name from _Michel Marcus_, Apr 27 2016
%E Errors in name and terms a(14), a(28), and a(56) corrected by _Daniel Zhu_, Feb 16 2024
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