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 A246547 Prime powers p^e where p is a prime and e >= 2 (prime powers without the primes or 1). 105
 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 Jens Kruse Andersen, Table of n, a(n) for n = 1..10000 Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024. 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 (Python) from sympy import primepi, integer_nthroot def A246547(n): def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))) kmin, kmax = 1, 2 while f(kmax) >= kmax: kmax <<= 1 while True: kmid = kmax+kmin>>1 if f(kmid) < kmid: kmax = kmid else: kmin = kmid if kmax-kmin <= 1: break return kmax # Chai Wah Wu, Aug 14 2024 CROSSREFS Essentially the same as A025475. Cf. A000961, A001597, A025528, A051953, A136141, A246655. 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 Adjacent sequences: A246544 A246545 A246546 * A246548 A246549 A246550 KEYWORD nonn,easy,changed AUTHOR Joerg Arndt, Aug 29 2014 STATUS approved

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Last modified September 17 11:42 EDT 2024. Contains 375987 sequences. (Running on oeis4.)