A257278 Prime powers p^m with p <= m.

4, 8, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 823543, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 5764801, 8388608, 9765625

Offset: 1

Author

M. F. Hasler, Apr 28 2015

Keywords

• nonn

Comments

Might be called "high" powers of primes. Motivated by challenges for which low powers of large primes provide somewhat trivial solutions, cf. A257279. The definition also avoids the question of the whether prime itself is to be considered as a prime power or not, cf. A000961 vs. A025475. In view of the condition p <= n, up to 10^10, only powers of the primes 2, 3, 5 and 7 (namely, less than 10) can occur.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000961, A025475, A257279.

Cf. A000040, A051674 (subsequence).

Subsequence of A122494 and A192135 (p < m, subsequence).

Cf. A257572, A257573.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a257278 n = a257278_list !! (n-1)
a257278_list = f (singleton (4, 2)) 27 (tail a000040_list) where
   f s pp ps@(p:ps'@(p':_))
     | qq > pp   = pp : f (insert (pp * p, p) s) (p' ^ p') ps'
     | otherwise = qq : f (insert (qq * q, q) s') pp ps
     where ((qq, q), s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2015

Mathematica

seq[lim_] := Module[{s = {}, p = 2}, While[p^p <= lim, AppendTo[s, p^Range[p, Log[p, lim]]]; p = NextPrime[p]]; Sort[Flatten[s]]]; seq[10^7] (* Amiram Eldar, Apr 14 2025 *)

PARI

L=List();lim=10;forprime(p=1,lim,for(n=p,lim*log(lim)\log(p),listput(L,p^n)));listsort(L);L

Formula

a(n) = A257572(n) ^ A257573(n). - Reinhard Zumkeller, May 01 2015

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^(p-1)*(p-1)) = 0.55595697220270661763... - Amiram Eldar, Oct 24 2020