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A257278
Prime powers p^m with p <= m.
5
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.
a(n) = A257572(n) ^ A257573(n). - Reinhard Zumkeller, May 01 2015
LINKS
FORMULA
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^(p-1)*(p-1)) = 0.55595697220270661763... - Amiram Eldar, Oct 24 2020
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. A000040, A051674 (subsequence).
Subsequence of A122494 and A192135 (p < m, subsequence).
Sequence in context: A380732 A380731 A122494 * A257279 A334151 A299894
KEYWORD
nonn
AUTHOR
M. F. Hasler, Apr 28 2015
STATUS
approved