A097764 Numbers of the form (kp)^p for prime p and k=1,2,3,....

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

Offset: 1

Author

T. D. Noe, Aug 24 2004

Keywords

• easy
• nice
• nonn

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^(j-1) j^j) p) ^ p. - Franklin T. Adams-Watters, Sep 13 2015

Links

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

Crossrefs

Cf. A084746 (least k such that n^k-k is prime).

Cf. A097792 (numbers of the form 4k^4 or (kp)^p).

Cf. A000040, A051674, A255134 (first differences).

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a097764 n = a097764_list !! (n-1)
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

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]

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