A036454 Prime powers with special exponents: q^(p-1) where p > 2 and q are prime numbers.

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

Offset

1,1

Comments

Composite numbers with a prime number of divisors.

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

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

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

a(n) = A283262(n)^2. - Amiram Eldar, Jul 04 2022

Sum_{n>=1} 1/a(n) = Sum_{k>=2} P(prime(k)-1) = 0.54756961912815344341..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

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-François 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

Intersection of A002808 and A009087.

Cf. A000005, A010051, A010553, A010055, A100995, A036450, A036452, A036459, A283262.

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