A039993 -id:A039993 - cp's OEIS frontend

A239196 A variant of primeval numbers A072857 where primes are counted with repetition as in A075053, not as in A039993.

1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1136, 1139, 1237, 1337, 1379, 10013, 10039, 10079, 10133, 10136, 10139, 10379, 12379, 13679, 100136, 100139, 100379, 101237, 102347, 102379, 103679

Offset: 1

M. F. Hasler, Mar 12 2014

Coincides with A072857 up to a(10) = 1079, but then this sequence lists two "intermediate" records 1136, 1139, before a(13) = 1237 = A072857(11).
"With repetition" means that primes are counted several times if they are obtained from different (not distinct) digits (e.g., 13 is obtained twice from 113), but not if they are obtained as different permutations of the same digits (e.g., p=11 is *not* counted twice even though it results as identical permutation and transposition (2,1) from the digits [1,1]).
The initial a(1)=1 has been chosen for consistency with A072857, it could be argued that it should not be there or listed as a(0).
See A239197 for the record values A075053(a(n)) reached for these numbers which are the indices of the records in A075053.

m=-1;for(k=1, 9e9, A075053(k)>m&&print1(k",")+m=A075053(k)) \\ Not very efficient; from 199, 1999, 19999 etc one could jump to the next larger power of 10. - M. F. Hasler, Mar 12 2014

A328516 a(n) = A039993(A179239(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, 3, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 1, 0, 0, 1, 0, 2, 0, 0, 1, 1, 1, 2, 1, 2, 0, 0, 2, 0, 0, 1, 1, 1, 3, 0, 0, 1, 0, 0, 0, 2, 1, 4, 2, 1, 2, 5, 0, 2, 1, 3, 7, 2, 3, 2, 4, 3, 4, 1, 5, 4, 4, 2, 6, 3, 3, 5, 6, 5, 7, 11

Offset: 0

David A. Corneth, Oct 18 2019

A179239 lists the least number that has its permutation of digits.

			a(95) = 11 as A039993(A179239(95)) = A179239(137) and the embedded primes in the permutations of the digits are the eleven primes {3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317}.

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A039993, A179239.

a(n) = A039993(A179239(n)).

A179239 Permutation classes of integers, each identified by its smallest member.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 77, 78, 79, 80, 88, 89, 90, 99, 100, 101, 102, 103

Offset: 0

Aaron Dunigan AtLee, Jul 04 2010

Let the "permutation set" of a positive integer n be the set of all integers formed by permuting the digits of n. Two integers are "permutationally congruent" if they generate the same permutation set. A "permutation class" is a set of all permutationally congruent integers. This sequence lists each permutation class, identified by its smallest member.
These are also the positive integers in order, omitting any d-digit number n if a previously listed d-digit number is a permutation of the digits of n.
Range of A328447: smallest representative of the equivalence class of all numbers having the same digits up to permutation. Equivalently: Numbers with digits in nondecreasing order, except that the smallest nonzero digit must precede the zero digits. This sequence is useful when considering functions which depend only on the digits of n, e.g., the number of primes contained in n, cf. A039993, A039999, A075053 and the records therein, A072857 (primeval numbers) and A076497, resp. A239196 and A239197, etc. - M. F. Hasler, Oct 18 2019

			The permutation set of 24 is {24, 42}, and this is the equivalence class modulo permutations of both of them, so 24 is listed, but 42 is not.
The permutation set of 30 is {3, 30}, but 3 is not in the same permutation class as 30 since 30 cannot be obtained by permuting digits of 3. Therefore 30 is listed separately from 3.
The numbers 89 and 98 are also permutationally congruent and form a permutation class, so only the smaller one is listed.

Aaron Dunigan AtLee and Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 2997 terms from Aaron Dunigan AtLee; prefix 0 by _Georg Fischer_, Oct 24 2019)

A variant of A009994.
Cf. A047726, A035927 (Number of distinct n-digit numbers up to permutations of digits).
Cf. A004186, A328447: largest & smallest representative of the class of n.

maxTerm = 103; (*maxTerm is the greatest term you wish to see*) permutationSet[n_Integer] := FromDigits /@ Permutations[IntegerDigits[n]]; permutationCongruentQ[x_Integer, y_Integer] := Sort[permutationSet[x]] == Sort[permutationSet[y]]; DeleteDuplicates[Range[maxTerm], permutationCongruentQ]
f[n_] := Block[{a = {0}, b = {DigitCount[0]}, i, w}, Do[w = DigitCount@ i; AppendTo[b, w]; If[! MemberQ[Most@ b, w], AppendTo[a, i]], {i, n}]; Rest@ a]; f@ 103 (* or faster: *)
Select[Range@ 103, LessEqual @@ IntegerDigits@ # || And[Take[IntegerDigits@ #, Last@ DigitCount@ # + 1] == Reverse@ Take[Sort@ IntegerDigits@ #, Last@ DigitCount@ # + 1], LessEqual @@ DeleteCases[IntegerDigits@ #, d_ /; d == 0]] &] (* Michael De Vlieger, Jul 14 2015 *)

is(n) = {my(d=digits(n),i); for(i=2,#d, if(d[i]!=0, d=vecextract(d,concat([1],vector(#d-i+1,j,i-1+j))); break));d==vecsort(d)||n/10^valuation(n,10)<10}
\\given an element n, in base b, find the next element from the sequence.
nxt(n,{b=10}) = {my(d = digits(n)); i = #d; while(i>0&&d[i]==b-1,i--); if(i>1, if(d[i]>0, d[i]++, d[i]=d[1];);for(j=i+1,#d,d[j]=d[i]), if(i==1, d[i]++;for(j=2,#d,d[j]=0), return(10^(#d))));sum(j=1,#d,d[j]*10^(#d-j))} \\ David A. Corneth, Apr 23 2016

select( is_A179239(n)={n==A328447(n)}, [0..200]) \\ M. F. Hasler, Oct 18 2019

from itertools import count, chain, islice
from sympy.utilities.iterables import combinations_with_replacement
def A179239_gen(): # generator of terms
    return chain((0,),(int(a+''.join(b)) for l in count(1) for a in '123456789' for b in combinations_with_replacement('0'+''.join(str(d) for d in range(int(a),10)),l-1)))
A179239_list = list(islice(A179239_gen(),31)) # Chai Wah Wu, Sep 13 2022

Prefixed with a(0) = 0 by M. F. Hasler, Oct 18 2019

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

A072857 Primeval numbers: numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits.

Original entry on oeis.org

1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, 100279, 100379, 101237, 102347, 102379, 103679, 123479, 1001237, 1002347, 1002379, 1003679, 1012349, 1012379, 1023457, 1023467, 1023479, 1234579, 1234679, 10012349

Offset: 1

Lekraj Beedassy, Jul 26 2002

RECORDS transform of A039993. - N. J. A. Sloane, Jan 25 2008. See A239196 and A239197 for the RECORDS transform of the closely related sequence A075053. - M. F. Hasler, Mar 12 2014
"73 is the largest integer with the property that all permutations of all of its substrings are primes." - M. Keith
Smallest monotonic increasing subsequence of A076449. - Lekraj Beedassy, Sep 23 2006
From M. F. Hasler, Oct 15 2019: (Start)
All terms > 37 start with leading digit 1 and have all other digits in nondecreasing order. The terms are smallest representatives of the class of numbers having the same digits, cf. A179239 and A328447 which both contain this as a subsequence.
The frequency of primes is roughly 50% for the displayed values, but appears to decrease. Can it be proved that the asymptotic density is zero?
Can we prove that there are infinitely many even terms? (Of the form 10...01..12345678?)
Can it be proved that there is no term that is a multiple of 3? (Or the contrary? Are there infinitely many?) (End)

			1379 is in the sequence because it is the smallest number whose digital permutations form a total of 31 primes, viz. 3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173, 9371.

J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), "1379's quite primeval, is it not?", pp. 318-321, Pour la Science, Paris 2000.

Giovanni Resta, Table of n, a(n) for n = 1..100
C. K. Caldwell, The Prime Glossary, primeval number
J. P. Delahaye, Primes Hunters, 1379 is very primeval (in French)
M. Keith, Integers containing many embedded primes
W. Schneider, Primeval Numbers
N. J. A. Sloane, Transforms
G. Villemin's Almanach of Numbers, Nombre Primeval de Mike Keith
Wikipedia, Primeval number

Cf. A039993, A075053, A076497, A239196, A239197.
A076449 gives a similar sequence.
Cf. A119535 (prime subsequence).

(*first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Length[ Select[ FromDigits /@ Flatten[ Permutations /@ Subsets[ IntegerDigits[ n]], 1], PrimeQ[ # ] &]]; d = -1; Do[ b = f[n]; If[b > d, Print[n]; d = b], {n, 2^20}] (* Robert G. Wilson v, Feb 12 2005 *)
Join[{1},DeleteDuplicates[Table[{n,Count[Union[FromDigits/@Flatten[Permutations[#]&/@Subsets[IntegerDigits[n]],1]],?PrimeQ]},{n,2,125000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]]] (* The program generates the first 30 terms of the sequence. *) (* _Harvey P. Dale, Nov 16 2024 *)

A072857_upto(num_digits,s=1,m=-1,L=List())={for(n=s,num_digits, my(u=10^(n-1)); forvec(v=vector(n-(n>2),i,[0,if(n>6,9*(i+1)\n,n>3,10-(n-i)\.6,7)]), m<A039993(u+fromdigits(v)) && m=A039993(listput(L,u+fromdigits(v))),1)); Vec(L)} \\ Optional 2nd and 3rd arg allow to extend a previous computation. - M. F. Hasler, Oct 15 2019

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

Edited, corrected and extended by Robert G. Wilson v, Nov 12 2002

Comment corrected by N. J. A. Sloane, Jan 25 2008

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

A076449 Least number whose digits can be used to form exactly n different primes (not necessarily using all digits).

Original entry on oeis.org

1, 2, 25, 13, 37, 107, 127, 113, 167, 1027, 179, 137, 1036, 1127, 1013, 1137, 1235, 1136, 1123, 1037, 1139, 1079, 10124, 10126, 1349, 1279, 1237, 3479, 10699, 1367, 10179, 1379, 10127, 10079, 10138, 10123, 10234, 10235, 10247, 10339, 10267

Offset: 0

Lekraj Beedassy, Nov 07 2002

Smallest m such that A039993(m) = n. - M. F. Hasler, Mar 08 2014

Mike Keith conjectures that a(n) always exists and reports that he has checked this for n <= 66. - N. J. A. Sloane, Jan 25 2008

			a(10) = 179 because 179 is the least number harboring ten primes (namely 7, 17, 19, 71, 79, 97, 179, 197, 719, 971).

Michael S. Branicky, Table of n, a(n) for n = 0..2702 (terms 1..457 from Robert G. Wilson v)
Michael S. Branicky, Python program for A076449, A072857, A076730, A134596
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, Nombre Primeval de Mike Keith

Cf. A075053, A072857 gives a similar sequence, A134596.

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Length[ Select[ FromDigits /@ Flatten[ Permutations /@ Subsets[ IntegerDigits[ n]], 1], PrimeQ[ # ] &]]; t = Table[0, {50}]; Do[ a = f[n]; If[a < 50 && t[[a + 1]] == 0, t[[a + 1]] = n], {n, 12500}]; t (* Robert G. Wilson v, Feb 12 2005 *)

A076449(n)=for(m=1,oo,A039993(m)==n&&return(m)) \\ Not very efficient. - M. F. Hasler, Mar 08 2014

Python
```
# see linked program
```

a(n) = min { m | A039993(m)=n } = min A039993^{-1}(n). - M. F. Hasler, Mar 08 2014

Edited by Robert G. Wilson v, Nov 24 2002
Keith link repaired by Charles R Greathouse IV, Aug 13 2009
Definition reworded by M. F. Hasler, Mar 08 2014
a(26) corrected by Robert G. Wilson v, Mar 12 2014

A076497 Number of primes corresponding to n-th primeval number A072857(n).

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, 31, 33, 35, 41, 53, 55, 60, 64, 89, 96, 106, 122, 153, 188, 248, 311, 349, 402, 421, 547, 705, 812, 906, 1098, 1162, 1268, 1662, 1738, 1953, 2418, 2920, 3133, 3457, 4483, 4517, 4917, 5174, 5953, 6552, 6799, 8938, 10219

Offset: 1

Lekraj Beedassy, Nov 08 2002

			a(3) = 3 because the primeval number A072857(3) = 13 can be used to create 3 prime numbers, namely 3, 13 and 31.
a(6) = 7 because the primeval number A072857(7) = 113 can be used to create 7 prime numbers, namely 3, 11, 13, 31, 113, 131 and 311. (The two primes 13 and 31 can be obtained in 2 ways, therefore A075053(113) = 9.)

Giovanni Resta, Table of n, a(n) for n = 1..100
C. K. Caldwell, The Prime Glossary, Primeval Number
M. Keith, Integers containing many embedded primes
G. Villemin's Almanach of Numbers, Nombre Primeval de Mike Keith
Wikipedia, Primeval number.

Cf. A072857, A077623, A076730.

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]]; d = -1; Do[ b = f[n]; If[b > d, Print[b]; d = b], {n, 1, 10^6}]

a(n) = A039993(A072857(n)). - M. F. Hasler, Mar 12 2014

Edited and extended by Robert G. Wilson v, Nov 12 2002
Links fixed by Charles R Greathouse IV, Aug 13 2009
a(40)-a(54) from Giovanni Resta, Nov 06 2013

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

cp's OEIS Frontend

A239196 A variant of primeval numbers A072857 where primes are counted with repetition as in A075053, not as in A039993.

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PARI

A328516 a(n) = A039993(A179239(n)).

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A179239 Permutation classes of integers, each identified by its smallest member.

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

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A072857 Primeval numbers: numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits.

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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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A076449 Least number whose digits can be used to form exactly n different primes (not necessarily using all digits).

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