A239196 -id:A239196 - cp's OEIS frontend

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

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

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

A134597 a(n) gives the maximal value of A075053(m) for any n-digit number m.

Original entry on oeis.org

1, 4, 11, 31, 106, 402, 1953

Offset: 1

N. J. A. Sloane, Jan 25 2008

In A075053(m), the primes obtained as permutations of digits of m are counted several times if they can be obtained in several different ways. See sequence A076730 which uses A039993 instead, i.e., counting only different primes. - M. F. Hasler, Mar 11 2014

The original data given for n = 3, 4, 5 was erroneously A007526(n). - Up to n = 6, a(n) = A076730(n), but the two will differ not later than for n = 10, where A134596(10) = 1123456789 gives a(10) >= 398100 = A075053(1123456789) > A039993(1123456789) = 362451 = A076730(10). The difference arises because each prime containing a single '1' will be counted twice by A075053, but only once by A039993. - M. F. Hasler, Oct 14 2019

			From _M. F. Hasler_, Oct 14 2019: (Start)
a(2) = 4 = A075053(37), because from 37 one can obtain the primes {3, 7, 37, 73}, and there is obviously no 2-digit number which could give more primes.
a(3) = 11 = A075053(137), because from 137 one can obtain the primes {3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317}, and no 3-digit number yields more.
a(4) = 31 = A075053(1379), because from 1379 one can obtain the 31 primes {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}, and no 4-digit number yields more.
a(5) = 106 = A075053(13679). a(6) = 402 = A075053(123479).
a(7) = 1953 = A075053(1234679). (End)

M. Keith, Integers containing many embedded primes

Cf. A075053, A039993, A134596.

Cf. A239196 for record indices of A075053, A239197 for associated record values.

A134597(n)={my(m=0);forvec(D=vector(n,i,[0,9]), vecsum(D)%3||next;m=max(A075053(fromdigits(D),D),m),1);m} \\ M. F. Hasler, Oct 14 2019

a(n) <= A007526(n), with equality iff n <= 2. [Keith]

a(n) = max { A075053(m); 10^(n-1) <= m < 10^n } >= A076730(n) = max { A039993(m); 10^(n-1) <= m < 10^n }. - M. F. Hasler, Mar 11 2014

Link fixed by Charles R Greathouse IV, Aug 13 2009
Definition corrected by M. F. Hasler, Mar 11 2014
Data corrected and extended by M. F. Hasler, Oct 14 2019

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A239197 The record values A075053 associated to the records (indices) listed in A239196.

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

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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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A134597 a(n) gives the maximal value of A075053(m) for any n-digit number m.

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