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

1,1

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

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

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 *)

Prog

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

Crossrefs

Cf. A000961, A025475, A257279.

Cf. A000040, A051674 (subsequence).

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

Cf. A257572, A257573.

Sequence in context: A380732 A380731 A122494 * A257279 A334151 A299894

Adjacent sequences: A257275 A257276 A257277 * A257279 A257280 A257281

Keyword

nonn

Author

M. F. Hasler, Apr 28 2015

Status

approved