|
%I #21 Nov 18 2023 06:28:34
%S 4,16,27,36,64,100,144,196,216,256,324,400,484,576,676,729,784,900,
%T 1024,1156,1296,1444,1600,1728,1764,1936,2116,2304,2500,2704,2916,
%U 3125,3136,3364,3375,3600,3844,4096,4356,4624,4900,5184,5476,5776,5832,6084,6400
%N Numbers of the form (kp)^p for prime p and k=1,2,3,....
%C The polynomial x^n - n is reducible over the integers for n in this sequence.
%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.
%C The representation (k*p)^p is generally not unique, e.g. a(120) = 46656 = (108*2)^2 = (12*3)^3. - _Reinhard Zumkeller_, Feb 14 2015
%C This is also numbers of the form (km)^m for any m > 1, not just primes. Let m be > 1; then m has a prime factor, so let m=pj, p a prime and j an integer > 0. Then (km)^m = (kpj)^pj = (k^j p^j j^j)^p = ((k^j p^(j-1) j^j) p) ^ p. - _Franklin T. Adams-Watters_, Sep 13 2015
%H T. D. Noe, <a href="/A097764/b097764.txt">Table of n, a(n) for n = 1..1000</a>
%H A. Schinzel, <a href="https://algo.inria.fr/seminars/sem92-93/schinzel.pdf">Problems and results on polynomials</a>.
%t nMax=10000; lst={}; n=1; While[p=Prime[n]; p^p<=nMax, k=1; While[(k*p)^p<=nMax, AppendTo[lst, (k*p)^p]; k++ ]; n++ ]; Union[lst]
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, insert)
%o a097764 n = a097764_list !! (n-1)
%o a097764_list = f 0 (singleton (4, 2, 2)) $
%o tail $ zip a051674_list a000040_list where
%o f m s ppps'@((pp, p) : ppps)
%o | pp < qq = f m (insert (pp, p, 2) s) ppps
%o | qq == m = f m (insert ((k * q) ^ q, q, k + 1) s') ppps'
%o | otherwise = qq : f qq (insert ((k * q) ^ q, q, k + 1) s') ppps'
%o where ((qq, q, k), s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Feb 14 2015
%o (PARI) is(n)=my(b,e=ispower(n,,&b),f); if(e==0, return(0)); f=factor(e)[,1]; for(i=1,#f, if(b%f[i]==0, return(1))); 0 \\ _Charles R Greathouse IV_, Aug 29 2016
%Y Cf. A084746 (least k such that n^k-k is prime).
%Y Cf. A097792 (numbers of the form 4k^4 or (kp)^p).
%Y Cf. A000040, A051674, A255134 (first differences).
%K easy,nice,nonn
%O 1,1
%A _T. D. Noe_, Aug 24 2004
|
