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Related to n-th powers of polynomials.
(Formerly M0183)
4

%I M0183 #46 Mar 25 2024 06:44:00

%S 1,1,1,2,1,12,1,2,3,10,1,12,1,14,15,2,1,12,1,10,21,22,1,12,5,26,3,14,

%T 1,60,1,2,33,34,35,12,1,38,39,10,1,84,1,22,15,46,1,12,7,10,51,26,1,12,

%U 55,14,57,58,1,60,1,62,21,2,65,132,1,34,69,70,1,12,1,74,15,38,77,156,1

%N Related to n-th powers of polynomials.

%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 Robert Israel, <a href="/A005730/b005730.txt">Table of n, a(n) for n = 1..10000</a> (first 150 terms from Daniel Zhu).

%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) = A005729(n)/A005731(n).

%F a(n) = 1 if n is prime, a(n) = 2*s(n) if n is divisible by 6, a(n) = s(n) otherwise, where s(n) is the squarefree kernel of n (A007947); i.e., s(1)=1 and if n = Product(p_i^(e_i)) then s(n) = Product(p_i) [From Zhu]. - _Sean A. Irvine_, Aug 18 2016 [Corrected by _Daniel Zhu_, Feb 16 2024]

%p A7947:= proc(n) convert(numtheory:-factorset(n),`*`) end:

%p f:= proc(n) if isprime(n) then 1 elif n mod 6 = 0 then 2*A7947(n) else A7947(n) fi end proc:

%p map(f, [$1..100]); # _Robert Israel_, Mar 10 2024

%t a[n_] := If[PrimeQ[n], 1, With[{s = Times @@ FactorInteger[n][[All, 1]]}, If[Mod[n, 6] == 0, 2s, s]]];

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 25 2024 *)

%o (Python)

%o from math import prod

%o from sympy import isprime, primefactors

%o def A005730(n): return 1 if isprime(n) else prod(primefactors(n))<<(not n%6) # _Chai Wah Wu_, Mar 10 2024

%Y Cf. A005729, A005731.

%K nonn

%O 1,4

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Jan 24 2005

%E Incorrect terms corrected by _Daniel Zhu_, Feb 16 2024