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A036454 Prime powers with special exponents: q^(p-1) where p > 2 and q are prime numbers. 7
4, 9, 16, 25, 49, 64, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Composite numbers with a prime number of divisors.

(1 - A010051(a(n))) * A010055(a(n)) * A010051(A100995(a(n))+1) = 1. - Reinhard Zumkeller, Jun 05 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

d(d(a(n))) = 2, where d(x) = tau(x) = sigma_0(x) is the number of divisors of x.

a(n) = (n log n)^2 + 2n^2 log n log log n + O(n^2 log n). - Charles R Greathouse IV, Apr 26 2012

A036459(a(n)) = 2. - Ivan Neretin, Jan 25 2016

EXAMPLE

From powers of 2: 4,16,64,1024,4096,65536 are in the sequence since exponent+1 is also prime. The same powers of any prime base are also included.

MAPLE

N:= 10^5:

P1:= select(isprime, [2, seq(2*i+1, i=1..floor((sqrt(N)-1)/2))]):

P2:= select(`<=`, P1, 1+ilog2(N))[2..-1]:

S:= {seq(seq(p^(q-1), q = select(`<=`, P2, 1+floor(log[p](N)))), p=P1)}:

sort(convert(S, list)); # Robert Israel, May 18 2015

MATHEMATICA

specialPrimePowerQ[n_] := With[{f = FactorInteger[n]}, Length[f] == 1 && PrimeQ[f[[1, 1]]] && f[[1, 2]] > 1 && PrimeQ[f[[1, 2]] + 1]]; Select[Range[20000], specialPrimePowerQ]  (* Jean-Fran├žois Alcover, Jul 02 2013 *)

Select[Range[20000], ! PrimeQ[#] && PrimeQ[DivisorSigma[0, #]] &] (* Carlos Eduardo Olivieri, May 18 2015 *)

PROG

(PARI) for(n=1, 34000, if(isprime(n), n++, x=numdiv(n); if(isprime(x), print(n))))

(PARI) list(lim)=my(v=List(), t); lim=lim\1+.5; forprime(p=3, log(lim)\log(2) +1, t=p-1; forprime(q=2, lim^(1/t), listput(v, q^t))); vecsort(Vec(v))

\\ Charles R Greathouse IV, Apr 26 2012

(Haskell)

a009087 n = a009087_list !! (n-1)

a009087_list = filter ((== 1) . a010051 . (+ 1) . a100995) a000961_list

-- Reinhard Zumkeller, Jun 05 2013

(MAGMA) [n: n in [1..20000] | not IsPrime(n) and IsPrime(DivisorSigma(0, n))]; // Vincenzo Librandi, May 19 2015

CROSSREFS

Cf. A000005, A036450, A036452, A010553, A009087.

Cf. A002808.

Sequence in context: A075494 A063735 A056798 * A115648 A082522 A279096

Adjacent sequences:  A036451 A036452 A036453 * A036455 A036456 A036457

KEYWORD

nonn,easy,nice

AUTHOR

Labos Elemer

STATUS

approved

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Last modified April 20 22:22 EDT 2019. Contains 322310 sequences. (Running on oeis4.)