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A prime is weak if the closest other prime is smaller than it. If there is a tie the prime is not weak.

For example 73 is a weak prime because 71 is prime but 75 is composite.

Task

Write some computer code that when given a prime greater than 2 as input will determine if it is a weak prime. This is a standard so you should output two unique values for each of the two cases (e.g. weak and not weak).

This is so standard rules for the tag apply.

OEIS

Here are the first 47 weak primes:

3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643, 647

Here is the OEIS for weak primes (should return weak) OEIS A051635

Here is the OEIS for balanced primes (should return not weak) OEIS A006562

Here is the OEIS for strong primes (should return not weak) OEIS A051634

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  • \$\begingroup\$ not weak or strong? \$\endgroup\$ – CalculatorFeline Jul 8 '17 at 16:39
  • 7
    \$\begingroup\$ @CalculatorFeline not weak is different than strong \$\endgroup\$ – Wheat Wizard Jul 8 '17 at 16:39

22 Answers 22

26
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Jelly, 7 bytes

Æn+Æp>Ḥ

Try it online!

Explanation

           See if
Æn         the next prime
  +Æp      plus the previous prime
     >Ḥ    is greater than 2n

As a bonus, changing > to = or < checks for balanced and strong primes, respectively.

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  • \$\begingroup\$ That should be >, no? \$\endgroup\$ – Dennis Jul 8 '17 at 16:55
  • 2
    \$\begingroup\$ Oh wow, that's clever... \$\endgroup\$ – ETHproductions Jul 8 '17 at 16:55
  • \$\begingroup\$ I just worked this way out too. Nice work! \$\endgroup\$ – Jonathan Allan Jul 8 '17 at 16:59
  • \$\begingroup\$ This is so clever... \$\endgroup\$ – Erik the Outgolfer Jul 8 '17 at 17:06
12
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Mathematica, 24 bytes

n=NextPrime;2#+n@-#<n@#&

The NextPrime built-in can be (ab?)used to compute the previous prime by feeding it a negative argument.

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6
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Jelly, 9 bytes

ḤÆRạÞ⁸ḊḢ>

Returns 1 for weak and 0 for not weak or balanced (returns 1 for an input of 2)

Try it online!

How?

ḤÆRạÞ⁸ḊḢ> - Link: prime number > 2, p
Ḥ         - double -> 2*p
 ÆR       - yield primes between 2 and 2*p inclusive
     ⁸    - chain's left argument, p
    Þ     - sort by:
   ạ      -   absolute difference (i.e. distance from p)
      Ḋ   - dequeue (removes p from the list, since it has distance zero)
       Ḣ  - head (gives us the nearest, if two the smallest of the two)
        > - greater than p?
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  • \$\begingroup\$ Ninja'd me with a complex solution... \$\endgroup\$ – Erik the Outgolfer Jul 8 '17 at 16:50
  • \$\begingroup\$ It was a split-second! \$\endgroup\$ – Jonathan Allan Jul 8 '17 at 16:51
  • 1
    \$\begingroup\$ No it wasn't, it was 9 full seconds iirc. No, 10 seconds. \$\endgroup\$ – Erik the Outgolfer Jul 8 '17 at 16:51
  • \$\begingroup\$ So it was (looking at the times) it happened as I submitted here :) \$\endgroup\$ – Jonathan Allan Jul 8 '17 at 16:52
  • 1
    \$\begingroup\$ Well, it seems you just golfed faster than me...(it's quite a trip to first go from IIṠ⁼1 to II>0 to I<\)...yours is much different though. It seems you think differently than me...EDIT: Pietu1998 returned! \$\endgroup\$ – Erik the Outgolfer Jul 8 '17 at 16:53
5
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PHP, 69 bytes

prints one for weak prime and nothing for not weak prime

for(;!$t;$t=$d<2)for($n=$d=$argn+$i=-$i+$w^=1;$n%--$d;);echo$n<$argn;

Try it online!

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3
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Octave, 93 84 bytes

Thanks to @LuisMendo and @rahnema1 for saving bytes!

function r=f(x);i=j=x;do--i;until(i<1|isprime(i));do++j;until(isprime(j));r=x-i<j-x;

Try it online!

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  • \$\begingroup\$ Can't you use i-=1 etc? Also, end is not needed in the function; you can move it to the footer \$\endgroup\$ – Luis Mendo Jul 8 '17 at 17:10
3
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Maxima, 42 bytes

f(n):=is(n-prev_prime(n)<next_prime(n)-n);

Try it online!

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3
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MATL, 13 bytes

qZq0)G_Yq+GE>

This outputs 1 if weak, 0 otherwise.

Try it online!

Explanation

q      % Implicit input, Subtract 1
Zq     % Vector of primes up to that
0)     % Get last one
G      % Push input again
_Yq    % Next prime
+      % Add
G      % Push input
E      % Multiply by 2
>      % Greater than? Implicit display
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3
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GNU APL 1.2, 78 bytes

∇f N
X←(R←(~R∊R∘.×R)/R←1↓⍳N×2)⍳N
(|R[X-1]-N)<|R[X+1]-N
∇

∇f N declares a function that takes an argument.

(~R∊R∘.×R)/R←1↓⍳N×2 gives a list of all the primes from 2 to twice the argument. I'm assuming that the next prime is less than twice the original. If this is untrue, N*2 gives N squared and takes the same number of bytes (hopefully that's big enough to exceed the next prime). (See Wikipedia's explanation for how the prime-finding works)

X←(R←(...))⍳N assigns that list to vector R (overwriting its previous contents), finds the index of the original prime N in that list, and then assigns that index to X.

|R[X-1]-N computes the difference between the previous prime (because R contains the primes, the X-1th element is the prime before N) and N and then takes the absolute value (APL operates right-to-left).

|R[X+1]-N does the same, but for the next prime.

(|R[X-1]-N)<|R[X+1]-N prints 1 if the previous prime is closer to the original than the next prime and 0 otherwise. Parentheses are needed for precedence.

ends the function.

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2
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Jelly, 9 bytes

Æp;;ÆnI</

Try it online!

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2
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Pyth, 15 bytes

>-fP_ThQfPT_tQy

Try it here.

Uses Pietu1998's algorithm.

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2
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Perl 6, 41 bytes

{[>] map ->\n{$_+n,*+n...&is-prime},1,-1}

Try it online!

$_ is the argument to the function. The mapping function -> \n { $_ + n, * + n ... &is-prime } takes a number n and returns a sequence of numbers $_ + n, $_ + 2*n, ... that ends when it reaches a prime number. Mapping this function over the two numbers 1 and -1 produces a sequence of two sequences; the first starts with $_ + 1 and ends with the first prime number greater than $_, and the second starts with $_ - 1 and ends with the first prime number less than $_. [>] reduces this two-element list with the greater-than operator, returning true if the first sequence is greater (ie, longer) than the second.

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2
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Python 2.7 - 120 bytes

from math import*
i=lambda x:factorial(x-1)%x==x-1
def f(n,c):return 1 if i(n-c)>i(n+c) else 0 if i(n+c)>0 else f(n,c+1)

Since python doesn't have a built-in is prime function, we can use Wilson's theorem to get a nice short prime checker. Wilson's theorem states that a number is prime if and only if (n-1)! is congruent to -1 mod(n). Hence the function i will return 1 if the number is prime and 0 if it's not. Following that the f function will determine if the next prime from that number occurs first when incremented down rather than incremented up. If neither of the incremented numbers prime, it's just recursively called again.

Some example I/O

f(3,1)
1
f(15,1)
0
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2
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Python 2, 122 108 103 94 92 bytes

def a(n):
 r=[2];x=2
 while r[-1]<=n:x+=1;r+=[x]*all(x%i for i in r)
 return sum(r[-3:])>3*n

Try it online!

Uses Pietu's idea... and then saved 28 bytes by golfing shorter prime list iterators; then 2 more by replacing -3*n>0 with >3*n (d'oh!)

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2
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Regex (most flavors), 47 bytes

^(?=(x*)(?!(x+)(\2\2x)+$)\1)x+(?!(xx+)\4+$)\1\1

Try it online!

Takes input in unary. Outputs a match for weak primes, no match for non-weak primes. Works in ECMAScript, Perl, PCRE, Python, Ruby.

Explanation:

Let N be the input, A the closest prime < N, and B the closest prime > N. The main difficulty of a regex approach to this challenge is that we can’t represent numbers greater than the input, like B. Instead, we find the smallest b such that 2b + 1 is prime and 2b + 1 > N, which ensures 2b + 1 = B.

(?=
  (x*)              # \1 = N - b, tail = b
  (?!(x+)(\2\2x)+$) # Assert 2b + 1 is prime
  \1                # Assert b ≥ \1 (and thus 2b + 1 > N)
)

Then, note that we don’t actually need to find A. As long as any prime < N is closer to N than B, N is a weak prime.

x+                  # tail iterates over integers < N
(?!(xx+)\4+$)       # assert tail is prime
\1\1                # assert tail ≥ 2 * \1 (and thus tail + B > 2N)
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1
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Octave, 53 bytes

@(n)diff(list_primes(h=nnz(primes(n))+1)(h-2:h),2)>0

Try it online!

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1
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JavaScript ES6, 162 154 bytes

8 bytes save based on Jörg Hülsermann's trick "print nothing in one case". No need to ?"Y":"N" after one<two

var isWeak=

a=>{p=[2];i=0;f=d=>{j=p[i];l:while(j++){for(x=0;p[x]*p[x]<=j;x++){if(j%p[x]==0){continue l}}return p[++i]=j}};while(p[i]<a+1){f()};return a*2<p[i]+p[i-2]}

[43,//true
53,//false
7901,//false
7907,//true
1299853,//true
1299869//false
].forEach(n=>{console.log(n,isWeak(n))})

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0
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05AB1E, 17 bytes

[<Dp#]¹[>Dp#]+¹·›

Try it online!

Uses Pietu1998's algorithm.

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0
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Python 3, 149 bytes

f=lambda n,k=1,P=1:n*[0]and P%k*[k]+f(n-P%k,k+1,P*k*k)
def a(n):p=f(n);return p.index(n)in filter(lambda i:p[i]-p[i-1]<p[i+1]-p[i],range(1,len(p)-1))

Try it online!

I'm using a prime generating function (top line) taken from this old stack exchange answer.

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0
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JavaScript, 98 bytes

let test = _=>(o.innerHTML=f(+prime.value))
let f= 

n=>{P=n=>{for(i=n,p=1;--i>1;)p=p&&n%i};a=b=n;for(p=0;!p;P(--a));for(p=0;!p;P(++b));return n-a<b-n}
Enter Prime: <input id="prime">
<button type="button" onclick="test()">test if weak</button>
<pre id="o"></pre>

Less Golphed

n=>{
   P=  // is a Prime greater than 1, result in p
       n=>{
           for(i=n,p=1;--i>1;)
               p=p&&n%i
       };

   a=b=n; // initialize lower and upper primes to n
   for(p=0;!p;P(--a)); // find lower,
   for(p=0;!p;P(++b)); // find upper,
   return n-a<b-n // is weak result
}

Note the test code does not check the input "prime" is actually a prime.

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0
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braingasm, 23 22 bytes

Prints 1 for weak primes and 0 for not weak.

;>0$+L[->+>2[>q[#:Q]]]

Walkthrough:

;                       Read a number to cell 0
 >0$+                   Go to cell 1 and copy the value of cell 0
     L                  Make the tape wrap around after cell 1
      [              ]  Loop:
       ->+>               Decrease cell 1 and increase cell 0
           2[       ]     Twice do:
             >              Go to the other cell
              q[   ]        If it's prime:
                #:Q         Print the current cell number and quit
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0
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Julia 0.6, 64 bytes

g(x,i)=0∉x%(2:x-1)?1:1+g(x+i,i);x->g(x,1)&(g(x-1,-1)<g(x+1,1))
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0
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Python 2, 81 bytes

n=input()
a=b=c=i=2;p=1
while b<n:
 p*=i;i+=1
 if p*p%i:a,b,c=b,c,i
print a+c>2*b

Try it online!

Uses Wilson's theorem for the primality test.

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