A097792 Numbers of the form 4k^4 or (kp)^p for prime p > 2 and k = 1, 2, 3, ....

4, 27, 64, 216, 324, 729, 1024, 1728, 2500, 3125, 3375, 5184, 5832, 9261, 9604, 13824, 16384, 19683, 26244, 27000, 35937, 40000, 46656, 58564, 59319, 74088, 82944, 91125, 100000, 110592, 114244, 132651, 153664, 157464, 185193, 202500, 216000

Offset: 1

Author

T. D. Noe, Aug 24 2004

Keywords

• nonn

Comments

A result of Vahlen shows that the polynomial x^n + n is reducible over the integers for n in this sequence and no other n.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• A. Schinzel, Problems and results on polynomials, Algorithms Seminar, INRIA, 1992-1993.
• K. T. Vahlen, Über reductible Binome, Acta Mathematica 19:1 (December 1895), pp. 195-198.

Crossrefs

Cf. A093324 (least k such that n^k+k is prime), A097764 (numbers of the form (kp)^p).

Cf. A072883, A239666, A303121, A303122.

Programs

Mathematica

nMax=500000; lst={}; k=1; While[4k^4<=nMax, AppendTo[lst, 4k^4]; k++ ]; n=2; While[p=Prime[n]; p^p<=nMax, k=1; While[(k*p)^p<=nMax, AppendTo[lst, (k*p)^p]; k++ ]; n++ ]; Union[lst]

PARI

upto(n) = {my(res = List()); for(i = 1, sqrtnint(n \ 4, 4), listput(res, 4*i^4) ); forprime(p = 3, log(n), pp = p^p; for(k = 1, sqrtnint(n \ pp, p), listput(res, pp * k ^ p); ) ); listsort(res); res } \\ David A. Corneth, Jan 12 2019

PARI

select( {is_A097792(n, p=0)= n%4==0 && ispower(n\4,4) || ((2 < p = ispower(n,,&n)) && if(isprime(p), n%p==0, foreach(factor(p)[,1], q, q%2 && n%q==0 && return(1))))}, [1..10^4]) \\ M. F. Hasler, Jul 07 2024

Python

from sympy import isprime, perfect_power, primefactors
def is_A097792(n):
    return n%4==0 and (perfect_power(n//4,[4]) or n==4) or (
        p := perfect_power(n)) and p[1] > 2 and (p[0]%p[1]==0 if isprime(p[1])
        else any(p[0]%q==0 for q in primefactors(p[1]) if q > 2))
# M. F. Hasler, Jul 07 2024

Formula

Is a(n) ~ c * n^3? - David A. Corneth, Jan 12 2019