A039993 Number of different primes embedded in n.

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 3, 1, 1, 1, 3, 0, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 2, 3, 1, 4, 2, 1, 0, 1, 1, 2, 0, 1, 0, 2, 0, 0, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 2, 0, 0, 1, 3, 2, 4, 2, 2, 2, 1, 1, 3, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 0, 3, 1, 0, 0, 2, 1, 4, 2, 1

Offset

1,13

Comments

a(n) counts (distinct) permuted subsequences of digits of n which denote primes.

Links

T. D. Noe, Table of n, a(n) for n=1..10000

C. K. Caldwell, The Prime Glossary, Primeval Number

J. P. Delahaye, Primes Hunters, 1379 is very primeval (in French)

Mike Keith, Integers containing many embedded primes

W. Schneider, Primeval Numbers

G. Villemin's Almanach of Numbers, Mike Keith's Primeval Number (in French).

Example

a(17) = 3 since we can obtain 7, 17 and 71. a(22) = 1, since we can get only one prime (in contrast, A075053(22) = 2).
a(1013) = 14 because the prime subsets derived from the digital permutations of 1013 are {3, 11, 13, 31, 101, 103, 113, 131, 311, 1013, 1031, 1103, 1301, 3011}.

Mathematica

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Block[{a = Drop[ Sort[ Subsets[ IntegerDigits[n]]], 1], b = c = {}, k = 1, l}, l = Length[a] + 1; While[k < l, b = Append[b, Permutations[ a[[k]] ]]; k++ ]; b = Union[ Flatten[b, 1]]; l = Length[b] + 1; k = 1; While[k < l, c = Append[c, FromDigits[ b[[k]] ]]; k++ ]; Count[ PrimeQ[ Union[c]], True]]; Table[ f[n], {n, 1, 105}]
Table[Count[Union[FromDigits/@(Flatten[Permutations/@Subsets[ IntegerDigits[ n]], 1])], _?PrimeQ], {n, 110}] (* Harvey P. Dale, Nov 29 2017 *)

Prog

(PARI) A039993(n)={my(S=[], D=vecsort(digits(n))); for(i=1, 2^#D-1, forperm(vecextract(D, i), p, isprime(fromdigits(Vec(p)))||next; S=setunion(S, [fromdigits(Vec(p))]))); #S} \\ To avoid duplicate scan of identical subsets of digits, one could skip the corresponding range of indices i when a binary pattern ...10... is detected. - M. F. Hasler, Mar 08 2014, simplified Oct 15 2019
(Python)
from itertools import permutations
from sympy import isprime
def a(n):
    l=list(str(n))
    L=[]
    for i in range(len(l)):
        L+=[int("".join(x)) for x in permutations(l, i + 1)]
    return len([i for i in set(L) if isprime(i)])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 25 2017
(Python)
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime
def A039993(n): return sum(1 for l in range(1, len(str(n))+1) for a in multiset_permutations(str(n), size=l) if a[0] !='0' and isprime(int(''.join(a)))) # Chai Wah Wu, Sep 13 2022

Crossrefs

Different from A075053. For records see A072857, A076497. See also A134596, A134597.

Cf. A039999.

Sequence in context: A325669 A332488 A068153 * A075053 A328516 A328517

Adjacent sequences: A039990 A039991 A039992 * A039994 A039995 A039996

Keyword

nonn,base

Author

David W. Wilson

Extensions

Edited by Robert G. Wilson v, Nov 25 2002

Keith link repaired by Charles R Greathouse IV, Aug 13 2009

Status

approved