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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)
4
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 (list; graph; refs; listen; history; text; internal format)
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.
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
Sequence in context: A284577 A122018 A363395 * A271504 A086660 A271503
KEYWORD
nonn,nice,easy
AUTHOR
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

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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)