A134597 -id:A134597 - cp's OEIS frontend

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

David W. Wilson

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

			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}.

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).

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

Cf. A039999.

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 *)

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

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

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

Edited by Robert G. Wilson v, Nov 25 2002

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

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

Original entry on oeis.org

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

David W. Wilson

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

			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.

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

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).

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

[ #[ 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

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

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

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

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

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

Edited by M. F. Hasler, Oct 14 2019

A075053 Number of primes (counted with repetition) that can be formed by rearranging some or all of the digits of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 12 2002

"Counted with repetition" means that if the same prime can be obtained using different digits, then it is counted several times (e.g., 13 obtained from 113 using the 1st and 3rd digit or the 2nd and 3rd digit), but not so if it is obtained as different permutations of the same digits (e.g., a(11) = 1 because the identical permutation and the transposition (2,1) of the digits [1,1] both yield 11, but this does not count twice since the same digits are used). - M. F. Hasler, Mar 12 2014

See A039993 for the variant which counts only distinct primes, e.g., A039993(22) = 1, A039993(113) = 7. See A039999 for the variant which requires all digits to be used. - M. F. Hasler, Oct 14 2019

			From 13 we can obtain 3, 13 and 31 so a(13) = 3. In the same way, a(17) = 3.
From 22 we obtain 2 in two ways, so a(22) = 2.
a(101) = 2 because from the digits of 101 one can form the primes 11 (using digits 1 & 3) and 101 (using all digits). The prime 11 = 011 formed using all 3 digits does not count separately because it has a leading zero.
a(111) = 3 because one can form the prime 11 using digits 1 & 2, 1 & 3 or 2 & 3.

T. D. Noe, Table of n, a(n) for n = 0..10000
M. F. Hasler, Primeval numbers, OEIS wiki, 2014, updated 2019.

Different from A039993 (counts only distinct primes). Cf. A072857, A076449.
Cf. A039999 (use all digits, count only distinct primes, allow leading zeros), A046810 (similar but don't allow leading zeros).
Cf. A134597 (maximum over all n-digit numbers), A076730 (maximum of A039993 for all n-digit numbers).
Cf. A239196 and A239197 for record indices and values.

f[n_] := Length@ Select[ Union[ FromDigits@# & /@ Flatten[ Subsets@# & /@ Permutations@ IntegerDigits@ n, 1]], PrimeQ@# &]; Array[f, 105, 0] (* Robert G. Wilson v, Mar 12 2014 *)

A075053(n,D=vecsort(digits(n)),S=0)=for(b=1,2^#D-1,forperm(vecextract(D,b),p,p[1]&&isprime(fromdigits(Vec(p)))&&S++));S \\ Second optional arg allows to give the digits if they are already known. - M. F. Hasler, Oct 14 2019, replacing earlier code from 2014.

from itertools import combinations
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime
def A075053(n): return sum(1 for l in range(1,len(str(n))+1) for a in combinations(str(n),r=l) for b in multiset_permutations(a) if b[0] !='0' and isprime(int(''.join(b)))) # Chai Wah Wu, Sep 13 2022

a(n) >= A039993(n) with equality iff n has no duplicate digit. - M. F. Hasler, Oct 14 2019

a(n) = Sum_{k in S(n)} A039999(k), if n has not more than one digit 0, where S(n) are the numbers whose nonzero digits are a subsequence of those of n, and which contain the digit 0 if n does, taken with multiplicity (for numbers using some but not all digits that are repeated in n). - M. F. Hasler, Oct 15 2019

Corrected and extended by John W. Layman, Oct 15 2002

Examples & crossrefs edited by M. F. Hasler, Oct 14 2019

A134596 The largest n-digit primeval number A072857.

Original entry on oeis.org

2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789

Offset: 1

N. J. A. Sloane, Jan 25 2008

Former definition: The least n-digit number m (i.e., m >= 10^(n-1)) which yields A076730(n) = the maximum, for m < 10^n, of A039993(m) = number of primes that can be formed using some or all digits of m.

Subsequence of A072857 consisting of the largest terms of given length. - M. F. Hasler, Mar 12 2014

M. Keith, Integers containing many embedded primes

Cf. A039993, A072857, A076730, A134597, A076449.

A134596(n,A=A072857)=vecmax(select(t->logint(t,10)+1==n,A)) \\ where A072857 must comprise all n digit terms of that sequence. - M. F. Hasler, Oct 14 2019

Python
```
# see linked program in A076449
```

a(n) = max { m in A072857, m < 10^n }. - M. F. Hasler, Mar 12 2014

Link fixed by Charles R Greathouse IV, Aug 13 2009

Definition reworded and values of a(6)-a(11) added by M. F. Hasler, Mar 11 2014

A076730 Maximum number of (distinct) primes that an n-digit number may shelter (i.e., primes contained among all digital substrings' permutations).

Original entry on oeis.org

1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505

Offset: 1

Lekraj Beedassy, Nov 08 2002

See sequence A134596 for the least numbers of given length which yields these maxima over n-digit indices for A039993. - M. F. Hasler, Mar 11 2014

By definition this is a subsequence of A076497. The term a(10) was incorrectly given as 398100 = A075053(1123456789), which double-counts each prime using only one digit '1'. But a(10) = A039993(1123456789) = A076497(80) = 362451. The values given for a(9) and a(11) were also incorrect, the latter probably for the same reason, and for a(9) probably due to double-counting of primes with leading zeros. - M. F. Hasler and David A. Corneth, Oct 15 2019

			We have a(3)=11, since among numbers 100 through 999, the smallest ones having 5, 6, 7, 8, 10, 11 embedded primes are respectively 107, 127, 113, 167, 179, 137 (the last of these being the first reaching the maximum number of 11 embedded primes, viz. 3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317).

M. Keith, Integers containing many embedded primes
W. Schneider, MATHEWS, Primeval Numbers

Cf. A072857, A076449, A076497, A134596 (largest n-digit primeval number).

Cf. A075053 (a variant of A039993), A134597 (= max A075053(1..10^n-1)).

a(n,m=0)=for(k=10^(n-1),10^n-1,A039993(k)>m&&m=A039993(k));m \\ M. F. Hasler, Mar 09 2014

Python
```
# see linked program in A076449
```

a(n) = A039993(A134596(n)) = max { A039993(m); m in A072857 and m < 10^n }. - M. F. Hasler, Mar 12 2014

a(n) = A076497(k) for k such that A072857(k) = A134596(n). - M. F. Hasler, Oct 15 2019

Link fixed by Charles R Greathouse IV, Aug 13 2009
a(6) from M. F. Hasler, Mar 09 2014
a(7)-a(11) from Robert G. Wilson v, Mar 11 2014
a(9)-a(11) corrected by M. F. Hasler, Oct 15 2019

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A039993 Number of different primes embedded in n.

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

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A075053 Number of primes (counted with repetition) that can be formed by rearranging some or all of the digits of n.

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A134596 The largest n-digit primeval number A072857.

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A076730 Maximum number of (distinct) primes that an n-digit number may shelter (i.e., primes contained among all digital substrings' permutations).

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