|
%I #23 Jan 14 2019 04:37:03
%S 4,27,64,216,324,729,1024,1728,2500,3125,3375,5184,5832,9261,9604,
%T 13824,16384,19683,26244,27000,35937,40000,46656,58564,59319,74088,
%U 82944,91125,100000,110592,114244,132651,153664,157464,185193,202500,216000
%N Numbers of the form 4k^4 or (kp)^p for prime p > 2 and k = 1, 2, 3, ....
%C 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.
%H David A. Corneth, <a href="/A097792/b097792.txt">Table of n, a(n) for n = 1..10000</a>
%H A. Schinzel, <a href="http://algo.inria.fr/seminars/sem92-93/schinzel.pdf">Problems and results on polynomials</a>, Algorithms Seminar, INRIA, 1992-1993.
%H K. T. Vahlen, <a href="http://dx.doi.org/10.1007/BF02402875">Über reductible Binome</a>, Acta Mathematica 19:1 (December 1895), pp. 195-198.
%F Is a(n) ~ c * n^3? - _David A. Corneth_, Jan 12 2019
%t 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]
%o (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
%Y Cf. A093324 (least k such that n^k+k is prime), A097764 (numbers of the form (kp)^p).
%Y Cf. A072883, A239666, A303121, A303122.
%K nonn
%O 1,1
%A _T. D. Noe_, Aug 24 2004
|
