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OEIS: A167171

A dense number is a number that has exactly as many prime divisors as non-prime divisors (including 1 and itself as divisors). Equivalently, it is either a prime or a product of two distinct primes. The first 100 dense numbers are:

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122, 123, 127, 129, 131, 133, 134, 137, 139, 141, 142, 143, 145, 146, 149, 151, 155, 157, 158, 159, 161, 163, 166, 167, 173, 177, 178, 179, 181, 183, 185, 187, 191, 193, 194

Given a nonnegative integer n, output dense(n). n may be 0-indexed or 1-indexed.

Reference implementation (Sage)

import itertools

def dense_numbers():
    n = 1
    while True:
        prime_divisors = [x for x in divisors(n) if x.is_prime()]
        non_prime_divisors = [x for x in divisors(n) if not x.is_prime()]
        if len(prime_divisors) == len(non_prime_divisors):
            yield n
        n += 1

N = 20
        
print itertools.islice(dense_numbers(), N, N+1).next()

Try it online

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  • \$\begingroup\$ So many prime number sequences... I didn't know so many existed \$\endgroup\$ – Beta Decay Aug 15 '16 at 20:37
  • 2
    \$\begingroup\$ @βετѧΛєҫαγ There are also primes called Sexy Primes ( ͡° ͜ʖ ͡°). \$\endgroup\$ – Adnan Aug 15 '16 at 20:39
  • \$\begingroup\$ @Adnan Oh myy ;D \$\endgroup\$ – Beta Decay Aug 15 '16 at 20:40
  • \$\begingroup\$ What is the maximum value for n? \$\endgroup\$ – R. Kap Aug 16 '16 at 17:18
  • \$\begingroup\$ @R.Kap As high as your language of choice can go. \$\endgroup\$ – user45941 Aug 16 '16 at 18:13
2
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Jelly, 9 bytes

ÆE²Sḍ2µ#Ṫ

Reads from STDIN and uses 1-based indexing. Try it online!

How it works

ÆE²Sḍ2µ#Ṫ  Main link. No arguments. Implicit argument: 0

      µ#   Read an integer n from STDIN and execute the chain to the left for
           k = 0, 1, 2, ... until n of them return a truthy value.
           Return the array of matches.
ÆE           Compute the exponents of k's prime factorization.
  ²          Square each exponent.
   S         Compute the sum of all squares.
    ḍ2       Test if 2 is divisible by the result (true iff  the sum is 1 or 2).
        Ṫ  Tail; extract the last (n-th) matching value of k.
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1
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05AB1E, 12 11 bytes

1-indexed

µ             # while counter != input
 NÑ           # get divisors of current number
   p          # check if prime
    D         # duplicate
     O        # sum one copy
      s_O     # invert and sum the other copy
         Q½   # if equal increase counter

Try it online

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1
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Brachylog, 17 bytes

:1yt.
1<.=$p#dl<3

Try it online!

Predicate 0 (main predicate)

:1yt.
:1y     Find the first (input) solutions of predicate 1
   t    Last element
    .   Unify with output

Predicate 1 (auxiliary predicate)

1<.=$p#dl<3
1<.            1 < output
  .=           assign a value to output
  . $p#d       output's prime factorization contains no duplicate
        l      and the length
         <3    is less than three
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1
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Actually, 12 bytes

All credits to Dennis for his algorithm.

`w♂N;*2%Y`╓N

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`w♂N;*2%Y`╓N

`        `     define a function
 w             prime factorization in exponent form:
               18 = (2^1)*(3^2) becomes [[2,1],[3,2]]
  ♂N           get the last element (exponent) of each sublist
    ;*         dot-product with self; equivalent to squaring
               each item and then taking the sum
      2%Y      test divisibility by 2
          ╓    first (input) solutions to the above function
           N   get the last element.
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1
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Husk, 13 bytes

!fö`¦2ṁo□LgpN

Try it online!

Explanation

!fö`¦2ṁo□LgpN
            N List of natural numbers
 fö           filtered by the following 4 functions:
          gp  prime factorization, grouped into equal runs
      ṁo□L    get length of each run, square and sum
   `¦2        is 2 divisible by the sum?
!             nth element in filtered list
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  • \$\begingroup\$ Divisible by 2 can just be less than 3 \$\endgroup\$ – Jo King Oct 5 '20 at 8:01
  • \$\begingroup\$ for some reason, <3 still requires an extra increment at the end, maintaining the same byte count. @JoKing \$\endgroup\$ – Razetime Oct 5 '20 at 8:06
  • \$\begingroup\$ ah, i guess <3 includes 1, which throws off the indexing \$\endgroup\$ – Jo King Oct 5 '20 at 8:14
0
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R, 93 Bytes

dense=function(n){a=b=0;for(i in which(!n%%1:n))if(which(!i%%2:i)+1==i)a=a+1 else b=b+1;a==b}

It has a tendency to throw a warning. Its not really a problem. Allowing the warning saves me 5 bytes.

Ungolfed

dense=function(n){
     a=b=0                                #Initializing
     factors = which(!n%%1:n)             #Finds all factors
     for(i in factors)                    #Loops through factors
         prime = which(!i%%2:i)+1==i      #Tests if current factor is prime. If it is -- the first term in this vector will be TRUE. Otherwise, it will be false.
           if (prime) a=a+1 else b=b+1    #If first term is true, add 1 to a. Else add one to b. 
      return(a==b)                        #Test equality of a and b.
}
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  • \$\begingroup\$ Can't you use the += operator to save 2 bytes? \$\endgroup\$ – R. Kap Aug 16 '16 at 17:21
  • \$\begingroup\$ Sadly, R doesn't have any useful incrementation operators like += or a++. Sometimes there can be shorter ways (taking advantage of loop structure mostly), but I don't know of one here. \$\endgroup\$ – user5957401 Aug 16 '16 at 17:41
0
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Python, 79 bytes

f=lambda n,k=2:n<1or-~f(n-(sum((k%i<1)+2*(k%i**2<1)for i in range(2,k))<3),k+1)

Uses 1-based indexing. Test it on Ideone.

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0
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PHP, 118 Bytes

for($i=1;!$o=$s[$argn];$r[2]?:$t+=2*$$i=1,$t?:$s[]=$i)for($t=0,$r=[],$n=++$i;$n;$n--)$i%$n?:$t+=${$r[]=$n}?:-1;echo$o;

Try it online!

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0
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Axiom, 102 bytes

f(n:PI):PI==(i:=1;repeat(i:=i+1;a:=divisors(i);2*#[x for x in a|prime?(x)]=#a=>(n=1=>break;n:=n-1));i)

ungolf and result

-- 1 base Indexed: return the n_th number i that has 2*#divisorsPrimeOf(i)=#divisors(i)
ff(n:PI):PI==
     i:=1
     repeat
        i:=i+1
        a:=divisors(i)
        2*#[x for x in a|prime?(x)]=#a=>(n=1=>break;n:=n-1)
     i

(3) -> [f(i)  for i in 1..23]
   (3)  [2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,31,33,34,35,37,38]
                                               Type: List PositiveInteger
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