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A039993 Number of different primes embedded in n. 15
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 (list; graph; refs; listen; history; text; internal format)
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

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, t, d); for(n=1, #D=vecsort(digits(n)), t=vector(n, i, 10^(i-1))~; forvec(i=vector(n, j, [1, #D]), d=vecextract(D, i); for(k=1, n!, isprime(p=vecextract(d, numtoperm(n, k))*t)&&S=setunion(S, Set(p))), 2)); #S} \\ For numbers with more than 6 digits it may be efficient to add X=[] and "#Set(i)<n && #X==(#X=setunion(X, Set(vecextract(D, i)*t)))&& next" inside forvec() to avoid scanning identical subsets of digits twice. But for smaller numbers this is rather slower. - M. F. Hasler, Mar 08 2014

(Python)

from itertools import permutations

from sympy import isprime

def a(n):

    l=list(str(n))

    L=[]

    for i in xrange(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 xrange(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: A226048 A325669 A068153 * A075053 A007362 A214709

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

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Last modified May 24 20:53 EDT 2019. Contains 323534 sequences. (Running on oeis4.)