A072857 -id:A072857 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A119535 Primeval primes: primes in A072857.

Original entry on oeis.org

2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079, 10139, 12379, 13679, 100279, 100379, 123479, 1001237, 1002347, 1003679, 1012379

Offset: 1

N. J. A. Sloane, Jan 25 2008

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
M. Keith, Integers containing many embedded primes

Cf. A028864, A072857.

More terms extracted from A072857. - R. J. Mathar, Jun 11 2010

A173052 Partial sums of A072857.

Original entry on oeis.org

1, 3, 16, 53, 160, 273, 410, 1423, 2460, 3539, 4776, 6143, 7522, 17601, 27724, 37860, 47999, 58236, 68515, 78882, 89261, 101640, 115319, 215598, 315977, 417214, 519561, 621940, 725619, 849098, 1850335, 2852682, 3855061, 4858740, 5871089

Offset: 1

Jonathan Vos Post, Feb 08 2010

Partial sums of primeval numbers. Primeval number: a prime which "contains" more primes in it than any preceding number. Here "contains" means may be constructed from a subset of its digits. E.g., 1379 contains 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 and 9371. The subsequence of prime partial sums of primeval numbers begins: 3, 53, 1423, 3539, 6143, 89261, 115319, 315977. What is the smallest primeval prime partial sums of primeval numbers, i.e. the intersection of this sequence with A119535?

			a(36) = 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.

Chris K. Caldwell AND G. L. Honaker, Jr., A BRIEF PRIME CURIOS! GLOSSARY.

Cf. A000040, A072857, A039993, A075053, A076497, A076449, A119535 (prime subsequence).

a(n) = SUM[i=1..n] A072857(i) = SUM[i=1..n] {numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits}.

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

A039993 Number of different primes embedded in n.

Original entry on oeis.org

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

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

cp's OEIS Frontend

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

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A134596 The largest n-digit primeval number A072857.

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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

A119535 Primeval primes: primes in A072857.

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A173052 Partial sums of A072857.

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

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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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