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A246547 Prime powers p^e where p is a prime and e >= 2 (prime powers without the primes or 1). 100
4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 8192, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These are sometimes called the proper prime powers.
A proper subset of A001597.
Equals A000961 \ A008578 = { x in A001597 | A001221(x)=1 }. - M. F. Hasler, Aug 29 2014
LINKS
FORMULA
a(n) = A025475(n+1). - M. F. Hasler, Aug 29 2014
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Dec 21 2020
MAPLE
isA246547 := proc(n)
local ifs;
ifs := ifactors(n)[2] ;
if nops(ifs) <> 1 then
false;
else
is(op(2, op(1, ifs)) > 1);
end if;
end proc:
for n from 2 do
if isA246547(n) then
print(n) ;
end if;
end do: # R. J. Mathar, Feb 01 2016 # Or:
isA246547 := n -> not isprime(n) and nops(numtheory:-factorset(n)) = 1:
select(isA246547, [$1..10000]); # Peter Luschny, May 01 2018
MATHEMATICA
With[{upto=15000}, Complement[Select[Range[upto], PrimePowerQ], Prime[ Range[ PrimePi[ upto]]]]] (* Harvey P. Dale, Nov 28 2014 *)
Select[ Range@ 15000, PrimePowerQ@# && !SquareFreeQ@# &] (* Robert G. Wilson v, Dec 01 2014 *)
With[{n = 15000}, Union@ Flatten@ Table[Array[p^# &, Floor@ Log[p, n] - 1, 2], {p, Prime@ Range@ PrimePi@ Sqrt@ n}] ] (* Michael De Vlieger, Jul 06 2018, faster program *)
PROG
(PARI) for(n=1, 10^5, if(isprimepower(n)>=2, print1(n, ", ")));
(PARI) m=10^5; v=[]; forprime(p=2, sqrtint(m), e=2; while(p^e<=m, v=concat(v, p^e); e++)); v=vecsort(v) \\ Faster program. Jens Kruse Andersen, Aug 29 2014
(SageMath)
def A246547List(n):
return [p for p in srange(2, n) if p.is_prime_power() and not p.is_prime()]
print(A246547List(14642)) # Peter Luschny, Sep 16 2023
CROSSREFS
Essentially the same as A025475.
There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018
Sequence in context: A134611 A134612 A025475 * A195942 A125643 A002760
KEYWORD
nonn,easy
AUTHOR
Joerg Arndt, Aug 29 2014
STATUS
approved

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Last modified March 18 22:50 EDT 2024. Contains 370951 sequences. (Running on oeis4.)