

A097764


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


15



4, 16, 27, 36, 64, 100, 144, 196, 216, 256, 324, 400, 484, 576, 676, 729, 784, 900, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3125, 3136, 3364, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832, 6084, 6400
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

The polynomial x^n  n is reducible over the integers for n in this sequence.
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.
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
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^(j1) j^j) p) ^ p.  Franklin T. AdamsWatters, Sep 13 2015


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Schinzel, Problems and results on polynomials


MATHEMATICA

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]


PROG

(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a097764 n = a097764_list !! (n1)
a097764_list = f 0 (singleton (4, 2, 2)) $
tail $ zip a051674_list a000040_list where
f m s ppps'@((pp, p) : ppps)
 pp < qq = f m (insert (pp, p, 2) s) ppps
 qq == m = f m (insert ((k * q) ^ q, q, k + 1) s') ppps'
 otherwise = qq : f qq (insert ((k * q) ^ q, q, k + 1) s') ppps'
where ((qq, q, k), s') = deleteFindMin s
 Reinhard Zumkeller, Feb 14 2015
(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


CROSSREFS

Cf. A084746 (least k such that n^kk is prime).
Cf. A097792 (numbers of the form 4k^4 or (kp)^p).
Cf. A000040, A051674, A255134 (first differences).
Sequence in context: A046346 A134330 A328415 * A227993 A072873 A072653
Adjacent sequences: A097761 A097762 A097763 * A097765 A097766 A097767


KEYWORD

easy,nice,nonn


AUTHOR

T. D. Noe, Aug 24 2004


STATUS

approved



