A039999 Number of permutations of digits of n which yield distinct primes.

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 3, 2, 0

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base
• nice

Comments

Consider all k! permutations of digits of a k-digit number n, discard initial zeros, count distinct primes.

Examples

			a(20) = 1, since from {02, 20} we get {2,20} and only 2 is prime.
From 107 we get 4 primes: (0)17, (0)71, 107 and 701; so a(107) = 4.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• C. Hilliard, PARI program.

Crossrefs

Cf. A046810.

Cf. A039993 (number of primes embedded in n), A076730 (maximum for n digits), A072857 (record indices: primeval numbers), A134596 (largest with n digits).

Cf. A075053 (as A039993 but counted with multiplicity), A134597 (maximum for n digits).

Programs

Haskell

import Data.List (permutations, nub)
a039999 n = length $ filter ((== 1) . a010051)
                   (map read (nub $ permutations $ show n) :: [Integer])
-- Reinhard Zumkeller, Feb 07 2011

Magma

[ #[ s: s in Seqset([ Seqint([m(p[i]):i in [1..#x] ], 10): p in Permutations(Seqset(x)) ]) | IsPrime(s) ] where m is map< x->y | [:i in [1..#x] ] > where x is [1..#y] where y is Intseq(n,10): n in [1..120] ]; // Klaus Brockhaus, Jun 15 2009

Mathematica

Table[Count[FromDigits/@Permutations[IntegerDigits[n]],?PrimeQ], {n,110}] (* _Harvey P. Dale, Jun 26 2011 *)

PARI

for(x=1, 400, print1(permprime(x), ",")) /* for definition of function permprime cf. link */ \\ Cino Hilliard, Jun 07 2009

PARI

A039999(n,D=vecsort(digits(n)),S)={forperm(D,p, isprime(fromdigits(Vec(p))) && S++);S} \\ Giving the 2nd arg avoids computing it and increases efficiency when the digits are already known. Must be sorted because forperm() only considers "larger" permutations. - M. F. Hasler, Oct 14 2019

Python

from sympy import isprime
from itertools import permutations
def a(n): return len(set(t for p in permutations(str(n)) if isprime(t:=int("".join(p)))))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Feb 17 2024

Extensions

Contribution of Cino Hilliard edited by Klaus Brockhaus, Jun 15 2009

Edited by M. F. Hasler, Oct 14 2019