login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A113877 Semiprimes to semiprime powers. 9
256, 1296, 4096, 6561, 10000, 38416, 46656, 50625, 194481, 234256, 262144, 390625, 456976, 531441, 1000000, 1048576, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 5764801, 6765201, 7529536, 9150625, 10077696, 10556001, 11316496, 11390625, 14776336 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is the semiprime analog of A053810 (numbers of the form p^q where p and q are primes).

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

FORMULA

{a(n)} = {a^b where a and b are elements of A001358}. {a(n)} = {(p*q)^(r*s) = (p^(r*s))*(q^r*s) for distinct primes p, q, r, s} UNION {(p*q)^(p*r) = (p^(p*r))*(q^(p*r)) for distinct primes p, q, r} UNION {(p*q)^(r*r) = (p^(r^2))*(q^(r^2)) for distinct primes p, q, r} UNION {(p*q)^(p*q)= (p^(p*q))*(q^(p*q)) for distinct primes p, q} UNION {(p^2)^(p^2) = p^(2*(p^2)) for prime p}.

a(n) ~ (n log n/log log n)^4. - Charles R Greathouse IV, Jun 05 2013

EXAMPLE

a(1) = 256 = 4^4 = semiprime(1)^semiprime(1).

a(2) = 1296 = 6^4 = semiprime(2)^semiprime(1).

a(3) = 4096 = 4^6 = semiprime(1)^semiprime(2).

a(4) = 6561 = 9^4 = semiprime(3)^semiprime(1).

a(5) = 10000 = 10^4.

MATHEMATICA

lim = 10^8; s = Select[Range[lim^(1/4)], Total[Transpose[FactorInteger[#]][[2]]] == 2 &]; t = {}; j = 1; While[b = s[[j]]; i = 1; While[a = s[[i]]; e = a^b; If[e <= lim, AppendTo[t, e]]; e < lim && i < Length[s], i++]; i > 1, j++]; t = Union[t] (* T. D. Noe, Jun 05 2013 *)

PROG

(PARI) is(n)=my(b, e=ispower(n, , &b), o); if(e==0, return(0)); o=bigomega(e); (o==2 && bigomega(b)==2) || (e%2==0 && o==3 && isprime(b)) \\ Charles R Greathouse IV, Jun 05 2013

(PARI) list(lim)=my(v=List()); for(e=4, log(lim\=1+.5)\log(4), if(bigomega(e)!=2, next); for(b=4, (lim+.5)^(1/e), if(bigomega(b)==2, listput(v, b^e)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 05 2013

CROSSREFS

Cf. A000040, A001358, A113854, A053810.

Sequence in context: A237952 A237235 A129539 * A237398 A237392 A205097

Adjacent sequences:  A113874 A113875 A113876 * A113878 A113879 A113880

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jan 27 2006

EXTENSIONS

Terms corrected by Charles R Greathouse IV, Jun 05 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 20 00:47 EDT 2019. Contains 324223 sequences. (Running on oeis4.)