Python 2, 42 bytes

f=lambda n,p=2:n%p and f(n,p+1)or p**n%n<1 

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Since Python doesn't have any built-ins for primes, we make do with checking divisibility.

We find the smallest prime p that's a factor of n by counting up p=2,3,4,... until n is divisible by p, that is n%p is zero. There, we check that this p is the only prime factor by checking that a high power of p is divisible by n. For this, p**n suffices.

As a program:

43 bytes

n=input() p=2 while n%p:p+=1 print p**n%n<1 

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This could be shorter with exit codes if those are allowed.

46 bytes

lambda n:all(n%p for p in range(2,n)if p**n%n) 

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Shakespeare Programming Language, 329 bytes

,.Ajax,.Page,.Act I:.Scene I:.[Enter Ajax and Page] Ajax:Listen tothy. Page:You cat. Scene V:. Page:You is the sum ofYou a cat. Is the remainder of the quotient betweenI you nicer zero?If soLet usScene V. Scene X:. Page:You is the cube ofYou.Is you worse I?If soLet usScene X. You is the remainder of the quotient betweenYou I.Open heart 

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Outputs 0 if the input is almost prime, and a positive integer otherwise. I am not sure this is an acceptable output; changing it would cost a few bytes.

Explanation:

• Scene I: Page takes in input (call this n). Initialize Ajax = 1.
• Scene V: Increment Ajax until Ajax is a divisor of Page; call the final value p This gives the smallest divisor of Page, which is guaranteed to be a prime.
• Scene X: Cube Ajax until you end up with a power of p, say p^k which is greater than n. Then n is almost-prime iff n divides p^k.

MATL, 4 bytes

Yf&= 

• For almost-primes the output is a matrix containing only 1s, which is truthy.
• Otherwise the output is a matrix containing several 1s and at least one 0, which is falsy.

Try it online! Or verify all test cases, including truthiness/falsihood test.

How it works

 % Implicit input Yf % Prime factors. Gives a vector with the possibly repeated prime factors &= % Matrix of all pair-wise equality comparisons % Implicit output 

05AB1E, 2 bytes

ÒË 

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Commented:

Ò -- Are all the primes in the prime decomposition Ë -- Equal? 

Wolfram Language (Mathematica), 11 bytes

PrimePowerQ 

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@Sisyphus saved 1 byte

R, 36 32 29 bytes

-3 bytes by outputting a vector of booleans without extracting the first element

!(a=2:(n=scan()))[!n%%a]^n%%n 

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Outputs a vector of booleans. In R, a vector of booleans is truthy iff the first element is TRUE.

First, find the smallest divisor p of n. We can do this by checking all integers (not only primes), as the smallest divisor of an integer (apart from 1) is always a prime number. Here, let a be all the integers between 2 and n, then p=a[!n%%a][1] is the first element of a which divides n.

Then n is almost prime iff n divides p^n.

This fails for any moderately large input, so here is the previous version which works for most larger inputs:

R, 36 33 bytes

!log(n<-scan(),(a=2:n)[!n%%a])%%1 

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Compute the logarithm of n in base p: this is an integer iff n is almost prime.

This will fail due to floating point inaccuracy for certain (but far from all) large-ish inputs, in particular for one test case: \$4913=17^3\$.

C (gcc), 43 bytes

f(n,i){for(i=1;n%++i;);n=i<n&&f(n/i)^i?:i;} 

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Returns p if n is almost-prime, and 1 otherwise.

f(n,i){ for(i=1;n%++i;); // identify i = the least prime factor of n n=i<n&&f(n/i)^i // if n is neither prime nor almost-prime ? // return 1 :i; // return i } 

APL (Dyalog Unicode), 7 bytes

⊢∊⍳∘.∧⍳ 

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Uses ⎕IO←0. Returns 0 if the input is almost-prime, 1 otherwise.

How it works

⊢∊⍳∘.∧⍳ ⍝ Input: integer n ≥ 2 ⍳∘.∧⍳ ⍝ Generate a table of LCMs over 0..n-1 ⊢∊ ⍝ Does the table contain n? 

The LCM of two numbers is equivalent to taking the maximum of powers in their prime factorizations. If n = p^k, any numbers smaller than n cannot have p^k in their prime factorization, so the LCM table does not contain n. Otherwise, we can find a pair of coprime numbers under n that multiplies to n, so the LCM table is guaranteed to contain at least one copy of n.

APL (Dyalog Classic), 33 31 26 bytes

{⍵∊∊(((⊢~∘.×⍨)1↓⍳)⍵)∘*¨⍳⍵} 

-5 bytes from Kevin Cruijssen's suggestion.

Warning: Very, very slow for larger numbers.

Explanation

{⍵∊∊(((⊢~∘.×⍨)1↓⍳)⍵)∘*¨⍳⍵} ⍵=n in all the following steps ⍳⍵ range from 1 to n ∘*¨ distribute power operator across left and right args (((⊢~∘.×⍨)1↓⍳)⍵) list of primes till n ∊ flatten the right arg(monadic ∊) ⍵∊ is n present in the primes^(1..n)? 

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J, ⁹ 8 bytes

1=#@=@q: 

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-1 byte thanks to xash

Tests if the self-classify = of the prime factors q: has length # equal to one 1=

Setanta, 61 59 bytes

gniomh(n){p:=2nuair-a n%p p+=1nuair-a n>1 n/=p toradh n==1} 

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Notes:

• The proper keyword is gníomh, but Setanta allows spelling it without the accents so I did so to shave off a byte.

Pyth, 4 bytes

!t{P 

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Explanation:

P - List of prime factors { - Remove duplicate elements t - Removes first element ! - Would return True if remaining list is empty, otherwise False 

-1 byte, thanks to hakr14

JavaScript (ES6), 43 bytes

Without BigInts

Returns a Boolean value.

f=(n,k=1)=>n%1?!~~n:f(n<0?n/k:n%++k?n:-n,k) 

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A recursive function that first looks for the smallest divisor \$k>1\$ of \$n\$ and then divides \$-n\$ by \$k\$ until it's not an integer anymore. (The only reason why we invert the sign of \$n\$ when \$k\$ is found is to distinguish between the two steps of the algorithm.)

If \$n\$ is almost-prime, the final result is \$-\dfrac{1}{k}>-1\$. So we end up with \$\lceil n\rceil=0\$.

If \$n\$ is not almost-prime, there exists some \$q>k\$ coprime with \$k\$ such that \$n=q\times k^{m}\$. In that case, the final result is \$-\dfrac{q}{k}<-1\$. So we end up with \$\lceil n\rceil<0\$.

JavaScript (ES11), 33 bytes

With BigInts

With BigInts, using @xnor's approach is probably the shortest way to go.

Returns a Boolean value.

f=(n,k=1n)=>n%++k?f(n,k):k**n%n<1 

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Regex (ECMAScript), 35 33 32 bytes

-2 bytes thanks to H.PWiz; the explanation for why this worked was more complicated
-1 bytes by returning to the original algorithm (with its simple explanation), by actually constructing values of \$a\$ not divisible by \$b\$ (32 bytes) instead of asserting \$a\$ is not divisible by \$b\$ (35 bytes); this is slower, but not exponentially so

^(?!(((x+)x+)(?=\2+$)\2*\3)\1+$) 

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The 35 byte version was written in 2014 by teukon and me.

This works by asserting that there do not exist any \$a\$ and \$b\$, both proper divisors of \$n\$ where \$a>b\$, for which \$a\$ is not divisible by \$b\$.

This will always be true when \$n\$ is of the form \$p^k\$ (a prime power), because we will have \$a=p^c\$ and \$b=p^d\$ where \$k > c > d \ge 0\$, thus \$a\$ will always be divisible by \$b\$.

But if \$n\$ is not of the form \$p^k\$, there will always be at least one counterexample \$a,b\$ such that \$a\$ is not divisible by \$b\$. Trivially we can choose two different prime factors of \$n\$, and they will not be mutually divisible.

This algorithm is similar to that used by Original Original Original VI's answer.

# For the purpose of these comments, the input number will be referred to as N. ^(?! # Assert that the following cannot match ( # Capture \1 to be the following: ((x+)x+) # Cycle through all values of \3 and \2 such that \2 > \3 > 1 (?=\2+$) # such that \2 is a proper divisor of N \2*\3 # Cycle through all values of \1 > \2 that aren't divisible # by \2, by letting \1 = \2 * A + \3 where A >= 0 ) \1+$ # where \1 is a proper divisor of N ) 

Bonus: Here is a 28 (27🐌) byte regex that matches numbers of the form \$p^k\$ where \$0 \le k \le 2\$:
^(?!((x+)x+)(?!\1*\2+$)\1+$), exponential slowdown 🐌 version: ^(?!((x+)+)(?!\1*\2+$)\1+$)
((x+)+) is equivalent to (x+)x* but exponentially slower)

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This is interesting because if I'd set out to implement that function intentionally, I would have come up with something like ^(?=(x?x+?)\1*$)((?=(x+)(\3+$))\4)?\1$ (38 bytes), which is 10 bytes longer.

It was inspired by the algorithm used by Bubbler's APL answer. The full prime power version of that would be:
^(?!((x+)(?=\2+$)x+)(?!\1*\2+$)\1+$) - ECMAScript, 36 bytes
^(?!((x+)x+)\B(?>\1*?(?<=^\2+))$) - .NET, 33 bytes
^(?!((x+)+)\B(?>\1*?(?<=^\2+))$) - .NET, 32 bytes, 🐌 exponential slowdown

And user41805's APL answer ported to regex is:
^(?!(xx+)\1*(?=\1*$)x(xx+)\2*$(?<=^\2+)) - ECMAScript 2018, 40 bytes
^(?!(?*(xx+)\1*$)(xx+)\2*(?=\2*$)x\1*$) - ECMAScript + (?*), 39 bytes

The algorithm used by 10+ of the other answers (that all the prime factors are the same / there is only one prime factor), ported to regex is:
^(?=(xx+?)\1*$)(?!(\1x+)(?<!^\3+(xx+))\2*$) - ECMAScript 2018, 43 bytes

Retina 0.8.2, 41 39 38 bytes

.+ $* ^(?!(((1+)1+)(?=\2+$)\2*\3)\1+$) 

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GAP 4.7, 31 bytes

n->Length(Set(FactorsInt(n)))<2

This is a lambda. For example, the statement

Filtered([2..81], n->Length(Set(FactorsInt(n)))<2 );

yields the list [ 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81 ].

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Factor, 35 bytes

: f ( n -- ? ) factors all-equal? ; 

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Haskell, 36 bytes

f n=mod(until((<1).mod n)(+1)2^n)n<1 

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36 bytes

f n=and[mod(gcd d n^n)n<2|d<-[1..n]] 

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39 bytes

f n=all((`elem`[1,n]).gcd n.(^n))[2..n] 

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39 bytes

f n=mod n(n-sum[1|1<-gcd n<$>[1..n]])<1 

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40 bytes

f n=and[mod(p^n)n<1|p<-[2..n],mod n p<1] 

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Io, 48 bytes

Port of @RobinRyder's R answer.

method(i,c :=2;while(i%c>0,c=c+1);i log(c)%1==0) 

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Explanation

method(i, // Take an input c := 2 // Set counter to 2 while(i%c>0, // While the input doesn't divide counter: c=c+1 // Increment counter ) i log(c)%1==0 // Is the decimal part of input log counter equal to 0? ) 

Assembly (MIPS, SPIM), 238 bytes, 6 * 23 = 138 assembled bytes

main:li$v0,5 syscall move$t3,$v0 li$a0,0 li$t2,2 w:bgt$t2,$t3,d div$t3,$t2 mfhi$t0 bnez$t0,e add$a0,$a0,1 s:div$t3,$t2 mfhi$t0 bnez$t0,e div$t3,$t3,$t2 b s e:add$t2,$t2,1 b w d:move$t0,$a0 li$a0,0 bne$t0,1,p add$a0,$a0,1 p:li$v0,1 syscall 

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Brachylog, 2 bytes

Are all prime factors equal?

ḋ= 

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APL (Dyalog Unicode) 17.0, 7 bytes

⍸2⌊/⊢∨⍳ 

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Uses a different approach from Bubbler's answer, but only works for Dyalog 17.X. Errors with a domain error on non-prime powers, and doesn't error on truthy inputs.

The train is decomposed as the following

┌─┬─────────────────┐ │⍸│┌─┬─────┬───────┐│ │ ││ │┌─┬─┐│┌─┬─┬─┐││ │ ││2││⌊│/│││⊢│∨│⍳│││ │ ││ │└─┴─┘│└─┴─┴─┘││ │ │└─┴─────┴───────┘│ └─┴─────────────────┘ 

First a range from 1 to the right argument is created using ⍳.

 (⍳) 10 1 2 3 4 5 6 7 8 9 10 

Then each value in this list is GCDed ∨ with the right argument ⊢.

 (⊢∨⍳) 10 1 2 1 2 5 2 1 2 1 10 

Then the pairwise minimum is taken, with the pairs overlapping, so 2⌊/1 3 2 is equivalent to (1⌊3)(3⌊2).

 (2⌊/⊢∨⍳) 10 1 1 1 2 2 1 1 1 1 

Monadic ⍸ in Dyalog 17.0 is a function that finds the indices of truthy values given a binary array, but this function of it is irrelevant for this solution. If the input is not a prime power, it will have at least one number greater than 1 when the above is performed (proof below). Otherwise if it is a prime power, the above will result in a list with only 1s. So applying ⍸ on a non-prime power will cause the function to error because its argument is not a binary array, whereas on a prime power it won't error because the argument would binary consisting only of 1s.

In Dyalog 18.0 ⍸ has been extended to also accept non-binary arrays, so the above won't work. Instead, using 'unique' ∪ in place of ⍸ will return the 1-length vector consisting only of 1 for truthy values, and a longer vector for falsey values.

Bubbler gave the following proof as to why 2⌊/⊢∨⍳ does not give a vector of 1s for falsey values.

Consider the Diophantine equation \$p_1a-p_2b=1\$, where \$p_1\$ and \$p_2\$ are distinct prime factors of a non-prime tower \$N\$ and \$a,b>0\$. This equation has solutions, a proof of which can be found in the following Wikipedia article. What this equation means is that you will have a situation where a multiple of \$p_1\$ occurs as the number immediately following a multiple of \$p_2\$. Since each prime factor divides the input \$N\$, their multiple, when GCDed with the input, will result in a value greater than 1. The fact that these multiples are less than \$N\$ is left as an exercise to the reader. So the two multiples of prime numbers following one another and that these multiples are less than \$N\$ mean that you have a situation where 2⌊/⊢∨⍳ will have a number greater than 1 in it.

This situation does not occur for prime powers because given any two prime factors \$p^m\$ and \$p^n\$ where \$n>m\$, there will never be a situation where \$p^na-p^mb=1\$ because that would imply \$p(p^{n-1}a-p^{m-1}b)=1\$, i.e. that \$p\$ divides 1 which is false.

Java, 69 64 bytes

n->{int f=0,t=1;for(;n%++t>0;);for(;n>1;n/=t)f|=n%t;return f<1;} 

-5 bytes thanks to @Deadcode.

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Explanation:

n->{ // Method with integer parameter and boolean return-type int f=0, // Flag-integer `f`, starting at 0 t=1; // Temp integer `t`, starting at 1 for(;n%++t>0;);// Increase `t` by 1 before every iteration with `++t`, // And continue looping until the input `n` is divisible by `t` for(;n>1; // Then start and continue a second loop as long as `n` is not 0 or 1: n/=t) // After every iteration: divide `n` by `t` f|= // Bitwise-OR the flag by: n%t; // `n` modulo-`t` return f<1;} // After both loops, return whether the flag is still 0 

If we would be allowed to ignore floating point inaccuracies, a port of @RobinRyder's R answer would be 64 bytes as well:

n->{int m=1;for(;n%++m>0;);return Math.log(n)/Math.log(m)%1==0;} 

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Explanation:

n->{ // Method with integer parameter and boolean return-type int m=1; // Minimum divisor integer `m`, starting at 1 for(;n%++m>0;); // Increase `m` by 1 before every iteration with `++m` // And continue looping until the input `n` is divisible by `m` return Math.log(n)/Math.log(m) // Calculate log_m(n) %1==0;} // And return whether it has no decimal values after the comma 

But unfortunately this approach fails for test case 4913 which would become 2.9999999999999996 instead of 3.0 due to floating point inaccuracies (it succeeds for all other test cases).
A potential fix would be 71 bytes:

n->{int m=1;for(;n%++m>0;);return(Math.log(n)/Math.log(m)+1e9)%1<1e-8;} 

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Retina 0.8.2, 50 43 bytes

.+ $* ^(?=(11+?)\1*$)((?=(1+)(\3+$))\4)*\1$ 

Try it online! Link includes faster test cases. Originally based on @Deadcode's answer to Match strings whose length is a fourth power, but saved 7 bytes thanks to @Deadcode by improving the basic algorithm. Explanation:

.+ $* 

Convert the input to unary.

^(?=(11+?)\1*$) 

Start by matching the smallest factor p of n. (p is necessarily prime, of course.)

(?=(1+)(\3+$)) 

Find ns largest proper factor m, which is equivalent to n divided by its smallest prime factor q (which will be equal to p on the first pass).

\4 

The factorisation also captures n-m, which is subtracted from the current value of n, effectively dividing n by q for the next pass of the loop.

(...)*\1$ 

Repeat this division by the smallest prime factor until n becomes p, which is only possible if n is a power of p.

MathGolf, 10 bytes

╒g¶mÉk╒#─╧ 

Port of @Razetime's APL (Dyalog Classic) answer, so make sure to upvote him as well!

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Explanation:

╒ # Push a list in the range [1, (implicit) input-integer) g # Filter it by: ¶ # Check if it's a prime m # Map each prime to, É # using the following three operations: k╒ # Push a list in the range [1, input-integer) again # # Take the current prime to the power of each value in this list ─ # After the map, flatten the list of lists ╧ # And check if this list contains the (implicit) input-integer # (after which the entire stack joined together is output implicitly) 

Japt, 6 bytes

I feel like this should be 1 or 2 bytes shorter ...

k ä¶ × 

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Jelly, 3 bytes

ÆfE 

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APL (Dyalog Unicode), 8 bytes

∨/1~⍨⊢∨⍳ 

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Outputs 1 if it isn't almost-prime, some other positive integer otherwise.

∨/1~⍨⊢∨⍳ ⊢∨⍳ ⍝ All divisors 1~⍨ ⍝ Remove 1 ∨/ ⍝ GCD 

Pyt, 4 bytes

ϼŁ1= 

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ϼ Implicit input; get unique ϼrime factors Ł Łength of array 1= equals 1?; implicit print 

Vyxal, 18 16 13 11 bytes

Thanks to The Thonnu for -7 bytes

ƛ?$Ḋ$æ∧;∑1= 

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Explanation:

ƛ ; # map over the input with this code: ? # get input $ # swap Ḋ # divisible? $ # swap æ # is the item prime? ∧ # logical and ∑ # sum 1= # is 1?