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

%I A257278 #24 Apr 14 2025 05:32:49
%S A257278 4,8,16,27,32,64,81,128,243,256,512,729,1024,2048,2187,3125,4096,6561,
%T A257278 8192,15625,16384,19683,32768,59049,65536,78125,131072,177147,262144,
%U A257278 390625,524288,531441,823543,1048576,1594323,1953125,2097152,4194304,4782969,5764801,8388608,9765625
%N A257278 Prime powers p^m with p <= m.
%C A257278 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.
%H A257278 Reinhard Zumkeller, <a href="/A257278/b257278.txt">Table of n, a(n) for n = 1..10000</a>
%F A257278 a(n) = A257572(n) ^ A257573(n). - _Reinhard Zumkeller_, May 01 2015
%F A257278 Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^(p-1)*(p-1)) = 0.55595697220270661763... - _Amiram Eldar_, Oct 24 2020
%t A257278 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 *)
%o A257278 (PARI) L=List();lim=10;forprime(p=1,lim,for(n=p,lim*log(lim)\log(p),listput(L,p^n)));listsort(L);L
%o A257278 (Haskell)
%o A257278 import Data.Set (singleton, deleteFindMin, insert)
%o A257278 a257278 n = a257278_list !! (n-1)
%o A257278 a257278_list = f (singleton (4, 2)) 27 (tail a000040_list) where
%o A257278    f s pp ps@(p:ps'@(p':_))
%o A257278      | qq > pp   = pp : f (insert (pp * p, p) s) (p' ^ p') ps'
%o A257278      | otherwise = qq : f (insert (qq * q, q) s') pp ps
%o A257278      where ((qq, q), s') = deleteFindMin s
%o A257278 -- _Reinhard Zumkeller_, May 01 2015
%Y A257278 Cf. A000961, A025475, A257279.
%Y A257278 Cf. A000040, A051674 (subsequence).
%Y A257278 Subsequence of A122494 and A192135 (p < m, subsequence).
%Y A257278 Cf. A257572, A257573.
%K A257278 nonn
%O A257278 1,1
%A A257278 _M. F. Hasler_, Apr 28 2015