A005730 Related to n-th powers of polynomials. (Formerly M0183)

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, 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, 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

Offset

1,4

Comments

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

Robert Israel, Table of n, a(n) for n = 1..10000 (first 150 terms from Daniel Zhu).

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

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]

Maple

A7947:= proc(n) convert(numtheory:-factorset(n), `*`) end:
f:= proc(n) if isprime(n) then 1 elif n mod 6 = 0 then 2*A7947(n) else A7947(n) fi end proc:
map(f, [$1..100]); # Robert Israel, Mar 10 2024

Mathematica

a[n_] := If[PrimeQ[n], 1, With[{s = Times @@ FactorInteger[n][[All, 1]]}, If[Mod[n, 6] == 0, 2s, s]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 25 2024 *)

Prog

(Python)
from math import prod
from sympy import isprime, primefactors
def A005730(n): return 1 if isprime(n) else prod(primefactors(n))<<(not n%6) # Chai Wah Wu, Mar 10 2024

Crossrefs

Cf. A005729, A005731.

Sequence in context: A069566 A336314 A066818 * A112284 A167401 A069249

Adjacent sequences: A005727 A005728 A005729 * A005731 A005732 A005733

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Jan 24 2005

Incorrect terms corrected by Daniel Zhu, Feb 16 2024

Status

approved