Additive primes
In mathematics, additive primes are prime numbers which sum of digits are also primes.
- Definitions
- Task
Write a program to determine (and show here) all additive primes whose elements are less than 500.
Optionally, show the number of additive primes.
- Also see
-
- the OEIS entry: A046704 additive primes.
- the prime-numbers entry: additive primes.
- the geeks for geeks entry: additive prime number.
- the prime-numbers fandom: additive primes.
APL
<lang APL>((+⌿(4/10)⊤P)∊P)/P←(~P∊P∘.×P)/P←1↓⍳500</lang>
- Output:
2 3 5 7 11 23 29 41 43 47 61 67 83 89 101 113 131 137 139 151 157 173 179 191 193 197 199 223 227 229 241 263 269 281 283
311 313 317 331 337 353 359 373 379 397 401 409 421 443 449 461 463 467 487
Factor
<lang factor>USING: formatting grouping io kernel math math.primes prettyprint sequences ;
- sum-digits ( n -- sum )
0 swap [ 10 /mod rot + swap ] until-zero ;
499 primes-upto [ sum-digits prime? ] filter [ 9 group simple-table. nl ] [ length "Found %d Erdős primes < 500.\n" printf ] bi</lang>
- Output:
2 3 5 7 11 23 29 41 43 47 61 67 83 89 101 113 131 137 139 151 157 173 179 191 193 197 199 223 227 229 241 263 269 281 283 311 313 317 331 337 353 359 373 379 397 401 409 421 443 449 461 463 467 487 Found 54 Erdős primes < 500.
Go
<lang go>package main
import "fmt"
func isPrime(n int) bool {
switch {
case n < 2:
return false
case n%2 == 0:
return n == 2
case n%3 == 0:
return n == 3
default:
d := 5
for d*d <= n {
if n%d == 0 {
return false
}
d += 2
if n%d == 0 {
return false
}
d += 4
}
return true
}
}
func sumDigits(n int) int {
sum := 0
for n > 0 {
sum += n % 10
n /= 10
}
return sum
}
func main() {
fmt.Println("Additive primes less than 500:")
i := 2
count := 0
for {
if isPrime(i) && isPrime(sumDigits(i)) {
count++
fmt.Printf("%3d ", i)
if count%10 == 0 {
fmt.Println()
}
}
if i > 2 {
i += 2
} else {
i++
}
if i > 499 {
break
}
}
fmt.Printf("\n\n%d additive primes found.\n", count)
}</lang>
- Output:
Additive primes less than 500: 2 3 5 7 11 23 29 41 43 47 61 67 83 89 101 113 131 137 139 151 157 173 179 191 193 197 199 223 227 229 241 263 269 281 283 311 313 317 331 337 353 359 373 379 397 401 409 421 443 449 461 463 467 487 54 additive primes found.
Julia
<lang julia>using Primes
let
p = primesmask(500)
println("Erdős primes under 500:")
pcount = 0
for i in 2:499
if p[i] && p[sum(digits(i))]
pcount += 1
print(lpad(i, 4), pcount % 20 == 0 ? "\n" : "")
end
end
println("\n\n$pcount Erdős primes found.")
end
</lang>
- Output:
Erdős primes under 500: 2 3 5 7 11 23 29 41 43 47 61 67 83 89 101 113 131 137 139 151 157 173 179 191 193 197 199 223 227 229 241 263 269 281 283 311 313 317 331 337 353 359 373 379 397 401 409 421 443 449 461 463 467 487 54 Erdős primes found.
Phix
<lang Phix>function erdos(integer p) return is_prime(sum(sq_sub(sprint(p),'0'))) end function sequence res = filter(get_primes_le(500),erdos) string r = join(shorten(apply(res,sprint),"",6)) printf(1,"%d additive primes found: %s\n",{length(res),r})</lang>
- Output:
54 additive primes found: 2 3 5 7 11 23 ... 443 449 461 463 467 487
REXX
<lang rexx>/*REXX program counts/displays the number of additive primes under a specified number N.*/ parse arg n cols . /*get optional number of primes to find*/ if n== | n=="," then n= 500 /*Not specified? Then assume default.*/ if cols== | cols=="," then cols= 10 /* " " " " " */ Ocols= cols; cols= abs(cols) /*Use the absolute value of cols. */ call genP n /*generate all primes under N. */ primes= 0 /*initialize number of additive primes.*/ $= /*a list of additive primes (so far). */
do j=2 until j>=n; if \!.j then iterate /*Is J not a prime? No, then skip it.*/
_= sumDigs(j); if \!._ then iterate /*is sum of J's digs a prime? No, skip.*/
primes= primes + 1 /*bump the count of additive primes. */
if Ocols<1 then iterate /*Build the list (to be shown later)? */
$= $ right(j, w) /*add the additive prime to the $ list.*/
if primes//cols\==0 then iterate /*have we populated a line of output? */
say substr($, 2); $= /*display what we have so far (cols). */
end /*j*/
if $\== then say substr($, 2) /*possible display some residual output*/ say say 'found ' primes " additive primes < " n exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sumDigs: parse arg x 1 s 2; do k=2 for length(x)-1; s= s + substr(x,k,1); end; return s /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: parse arg n; @.=.; @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13; @.7=17; #= 7
w= length(n); !.=0; !.2=1; !.3=1; !.5=1; !.7=1; !.11=1; !.13=1; !.17=1
do j=@.7+2 by 2 while j<n /*continue on with the next odd prime. */
parse var j -1 _ /*obtain the last digit of the J var.*/
if _ ==5 then iterate /*is this integer a multiple of five? */
if j // 3 ==0 then iterate /* " " " " " " three? */
/* [↓] divide by the primes. ___ */
do k=4 to # while k*k<=j /*divide J by other primes ≤ √ J */
if j//@.k == 0 then iterate j /*÷ by prev. prime? ¬prime ___ */
end /*k*/ /* [↑] only divide up to √ J */
#= # + 1; @.#= j; !.j= 1 /*bump prime count; assign prime & flag*/
end /*j*/
return</lang>
- output when using the default inputs:
2 3 5 7 11 23 29 41 43 47 61 67 83 89 101 113 131 137 139 151 157 173 179 191 193 197 199 223 227 229 241 263 269 281 283 311 313 317 331 337 353 359 373 379 397 401 409 421 443 449 461 463 467 487 found 54 additive primes < 500
Ring
<lang ring> load "stdlib.ring"
see "working..." + nl see "Erdős-primes are:" + nl
row = 0 limit = 500
for n = 1 to limit
num = 0
if isprime(n)
strn = string(n)
for m = 1 to len(strn)
num = num + number(strn[m])
next
if isprime(num)
row = row + 1
see "" + n + " "
if row%10 = 0
see nl
ok
ok
ok
next
see nl + "found " + row + " Erdös-primes." + nl see "done..." + nl </lang>
- Output:
working... Erdős-primes are: 2 3 5 7 11 23 29 41 43 47 61 67 83 89 101 113 131 137 139 151 157 173 179 191 193 197 199 223 227 229 241 263 269 281 283 311 313 317 331 337 353 359 373 379 397 401 409 421 443 449 461 463 467 487 found 54 Erdös-primes. done...
Wren
<lang ecmascript>import "/math" for Int import "/fmt" for Fmt
var sumDigits = Fn.new { |n|
var sum = 0
while (n > 0) {
sum = sum + (n % 10)
n = (n/10).floor
}
return sum
}
System.print("Additive primes less than 500:") var primes = Int.primeSieve(499) var count = 0 for (p in primes) {
if (Int.isPrime(sumDigits.call(p))) {
count = count + 1
Fmt.write("$3d ", p)
if (count % 10 == 0) System.print()
}
} System.print("\n\n%(count) additive primes found.")</lang>
- Output:
Additive primes less than 500: 2 3 5 7 11 23 29 41 43 47 61 67 83 89 101 113 131 137 139 151 157 173 179 191 193 197 199 223 227 229 241 263 269 281 283 311 313 317 331 337 353 359 373 379 397 401 409 421 443 449 461 463 467 487 54 additive primes found.