

A039993


Number of different primes embedded in n.


17



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
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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) [broken link]
M. 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^#D1, 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 list(permutations(l, i + 1))]
return len(list(filter(lambda i: isprime(i), list(set(L)))))
print [a(n) for n in range(1, 101)] # Indranil Ghosh, Jun 25 2017


CROSSREFS

Different from A075053. For records see A072857, A076497. See also A134596, A134597.
Cf. A039999.
Sequence in context: A329210 A325669 A068153 * A075053 A007362 A214709
Adjacent sequences: A039990 A039991 A039992 * A039994 A039995 A039996


KEYWORD

nonn,base,changed


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



