A084746 -id:A084746 - cp's OEIS frontend

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

T. D. Noe, Aug 24 2004

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

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

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).

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

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]

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

A099228 Primes of the form m^k-k, with m and k > 1.

Original entry on oeis.org

2, 5, 7, 23, 47, 61, 79, 167, 223, 359, 439, 503, 509, 727, 839, 997, 1019, 1087, 1223, 1367, 1847, 2207, 2399, 2741, 3023, 3719, 3967, 4093, 4759, 5039, 5623, 5927, 6553, 7919, 8179, 8647, 10607, 11447, 13687, 14159, 14639, 15619, 16127, 17159, 17573

Offset: 1

T. D. Noe, Oct 06 2004

nonn

It appears that primes of this form are much more common than primes of the form m^k+k (A099227).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A057897 (numbers of the form m^k-k, with m and k > 1), A084746 (least k such that n^k-k is prime).

Cf. A099227.

nLim=32000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Select[Union[lst], PrimeQ]

A084745 Smallest prime of the form n^k - k, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 3, 23, 5, 47, 7, 79, 997, 5559917313492231463, 11, 167, 13, 223, 4093, 24137563, 17, 359, 19, 439, 10947877107572929152919737180202022857988400441953615831

Offset: 2

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

nonn

a(11) > 379749833583227. Conjecture: No entry is zero.
a(23) is 150 digits long and too long to include. - Alec Mihailovs (Alec(AT)Mihailovs.com), Jun 16 2003
If n-1 is a prime then a(n)=n-1. - Farideh Firoozbakht, Aug 09 2014

			a(7) = 47 = 7^2 - 2.

Michel Marcus, Table of n, a(n) for n = 2..35

Cf. A084746.

a := proc(n) local k; k := 1; while not isprime(n^k-k) do k := k+1 od; n^k-k end: seq(a(n),n=2..35);

sp[n_]:=Module[{k=1},While[!PrimeQ[n^k-k],k++];n^k-k]; Array[sp,21,2] (* Harvey P. Dale, Jul 22 2021 *)

a(n)=my(k=1);while(!ispseudoprime(n^k-k),k++);return(n^k-k)
vector(20, n, a(n+1)) \\ Derek Orr, Aug 08 2014

a(n) = n^A084746(n) - A084746(n). - Michel Marcus, Aug 09 2014

More terms from Alec Mihailovs (Alec(AT)Mihailovs.com), Jun 16 2003

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

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A099228 Primes of the form m^k-k, with m and k > 1.

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A084745 Smallest prime of the form n^k - k, or 0 if no such prime exists.

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