A179239 -id:A179239 - cp's OEIS frontend

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

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)).

A365375 Numbers being the smallest positive integer having its digits (Cf. A179239) from which two digits can be chosen, the difference being any value from 0 to 9.

Original entry on oeis.org

100269, 100479, 101269, 101479, 102269, 102669, 102699, 104479, 104779, 104799, 200589, 202589, 205589, 205889, 205899, 300789, 303789, 307789, 307889, 307899, 1000269, 1000479, 1001269, 1001479, 1002269, 1002349, 1002359, 1002369, 1002379, 1002469, 1002479, 1002489, 1002569, 1002579

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Sep 02 2023

Anagrams of the terms are not included in the sequence.

There are 320 such numbers up to 10^7, the largest being 5067899.

			a(1) = 100269 and we have:
  0 = 0 - 0
  1 = 1 - 0
  2 = 2 - 0
  3 = 9 - 6
  4 = 6 - 2
  5 = 6 - 1
  6 = 6 - 0
  7 = 9 - 2
  8 = 9 - 1
  9 = 9 - 0
The integer 102069 being an anagram of 100269 is not in the sequence (though 102069 also produces the 10 digits).

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A179239, A219248.

lst={};Do[If[Union@Flatten[Abs@*Differences/@Subsets[IntegerDigits@k,{2}]]==Range[0,9],If[FreeQ[lst,s=Sort@IntegerDigits@k],AppendTo[lst,s];Print@k]],{k,10^6}]

from itertools import count, islice, combinations, combinations_with_replacement as mc
def c(t):
    d = list(map(int, t))
    return len(set(abs(d[i]-d[j]) for i, j in combinations(range(len(d)), 2))) == 10
def bgen():
    D = "123456789"
    return ((D[i],)+r for d in count(1) for i in range(9) for r in mc("0"+D[i:], d-1))
def agen():
    yield from (int("".join(t)) for t in filter(c, bgen()))
print(list(islice(agen(), 34))) # Michael S. Branicky, Sep 11 2024

Name specified by David A. Corneth, Sep 11 2024

A047726 Number of different numbers that are formed by permuting digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 6

Offset: 1

N. J. A. Sloane

The minimum value of a(A171102(n)) is 10*9!. - Altug Alkan, Jul 08 2016

			From 102 we get 102, 120, 210, 201, 12 and 21, so a(102)=6.
From 33950 with 5 digits, one '0', two '3', one '5' and one '9', we get 5! / (1! * 2! * 1! * 1!) = 60 different numbers and a(33950) = 60.  - _Bernard Schott_, Oct 20 2019

A. Dunigan AtLee, Table of n, a(n) for n = 1..100000.

Cf. A055098. Identical to A043537 and A043562 for n<100.

Cf. A179239. - Aaron Dunigan AtLee, Jul 14 2010

import Data.List (permutations, nub)
a047726 n = length $ nub $ permutations $ show n
-- Reinhard Zumkeller, Jul 26 2011

f:= proc(n) local L;
  L:= convert(n,base,10);
  nops(L)!/mul(numboccur(i,L)!,i=0..9);
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 08 2016

pd[n_]:=Module[{p=Permutations[IntegerDigits[n]]},Length[Union [FromDigits/@p]]]; pd/@Range[120]  (* Harvey P. Dale, Mar 22 2011 *)

a(n)=n=eval(Vec(Str(n)));(#n)!/prod(i=0,9,sum(j=1,#n,n[j]==i)!) \\ Charles R Greathouse IV, Sep 29 2011

A047726(n)={local(c=Vec(0,10)); apply(d->c[d+1]++, digits(n)); logint(n*10,10)!/prod(i=1,10,c[i]!)} \\ M. F. Hasler, Oct 18 2019

a(n) << n / (log_10 n)^4.5 by Stirling's approximation. - Charles R Greathouse IV, Sep 29 2011

a(n) = A000142(A055642(n))/Product_{k=0..9} A000142(A100910(n,k)). - Robert Israel, Jul 08 2016

Corrected by Henry Bottomley, Apr 19 2000

A101337 Sum of (each digit of n raised to the power (number of digits in n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 4, 5, 8, 13, 20, 29, 40, 53, 68, 85, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 25, 26, 29, 34, 41, 50, 61, 74, 89, 106, 36, 37, 40, 45, 52, 61, 72, 85, 100, 117, 49, 50, 53, 58, 65

Offset: 1

Gordon Hamilton, Dec 24 2004

Sometimes referred to as "narcissistic function" (in base 10). Fixed points are the narcissistic (or Armstrong, or plus perfect) numbers A005188. - M. F. Hasler, Nov 17 2019

			a(75) = 7^2 + 5^2 = 74 and a(705) = 7^3 + 0^3 + 5^3 = 468.
a(1.02e59 - 1) = 102429587095122578993551250282047487264694110769657513064859 ~ 1.024e59 is an example of n close to the limit beyond which a(n) < n for all n. - _M. F. Hasler_, Nov 17 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Wikipedia, Narcissistic number, as of Nov 18 2019.

Cf. A055642, A179239, A306360.

f:=func; [f(n):n in [1..75]]; // Marius A. Burtea, Nov 18 2019

Array[Total[IntegerDigits[#]^IntegerLength[#]]&,80] (* Harvey P. Dale, Aug 27 2011 *)

a(n)=my(d=digits(n)); sum(i=1,#d, d[i]^#d) \\ Charles R Greathouse IV, Aug 10 2017

apply( A101337(n)=vecsum([d^#n|d<-n=digits(n)]), [0..99]) \\ M. F. Hasler, Nov 17 2019

def A101337(n):
    s = str(n)
    l = len(s)
    return sum(int(d)**l for d in s) # Chai Wah Wu, Feb 26 2019

a(n) <= A055642(n)*9^A055642(n) with equality for all n = 10^k - 1. Write n = 10^x to get a(n) < n when 1+log_10(x+1) < (x+1)(1-log_10(9)) <=> x > 59.85. It appears that a(n) < n already for all n > 1.02*10^59. - M. F. Hasler, Nov 17 2019

Name changed by Axel Harvey, Dec 26 2011

Edited by M. F. Hasler, Nov 17 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

A099627 Triangle read by rows: T(n,k) = 2^n + 2^k - 1 with n >= k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19, 23, 31, 32, 33, 35, 39, 47, 63, 64, 65, 67, 71, 79, 95, 127, 128, 129, 131, 135, 143, 159, 191, 255, 256, 257, 259, 263, 271, 287, 319, 383, 511, 512, 513, 515, 519, 527, 543, 575, 639, 767, 1023, 1024, 1025, 1027, 1031, 1039

Offset: 0

Henry Bottomley, Oct 25 2004

Positive integers m where m-th Catalan number A000108(m) = C(2m,m)/(m+1) is not divisible by 4, i.e. where A048881(m) is 0 or 1.
Numbers in A000225 or A099628.
From Charles L. Hohn, Jul 25 2024: (Start)
Integers >=1 whose binary digit counts (number of 0s and number of 1s) are distinct from those of any smaller number.
Binary analog of A179239 for n>=1.
All integers whose binary expression conforms to regex /^10*1*$/, shown in base 10 in ascending numeric order. (End)
Together with 0 all fixed points of A073137. - Alois P. Heinz, Jan 30 2025

			Triangle starts:                  In binary:
   k = 0  1  2  3  4  5
n
0      1                               1
1      2  3                           10     11
2      4  5  7                       100    101    111
3      8  9 11 15                   1000   1001   1011   1111
4     16 17 19 23 31               10000  10001  10011  10111  11111
5     32 33 35 39 47 63           100000 100001 100011 100111 101111 111111
E.g. T(5,3) = 2^5 + 2^3-1 = 32 + 7 = 39 (100111 in binary).

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

A053221 (row sums), A000079 (left diagonal), A000225 (right diagonal).
A048645 (see formula).
Partial sums of A232089.
Cf. A073137, A179239.

a099627 n k = a099627_tabl !! n !! k
a099627_row n = a099627_tabl !! n
a099627_tabl = iterate (\xs@(x:_) -> (2 * x) : map ((+ 1) . (* 2)) xs) [1]
-- Reinhard Zumkeller, Dec 19 2012

Table[2^n+2^k -1,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 27 2016 *)

As sequence, a(n) = A048645(n+2) - 1.

G.f.: (1 - x - x^2*y)/((1 - x)*(1 - 2*x)*(1 - x*y)*(1 - 2*x*y)). - Stefano Spezia, Aug 11 2024

A293869 Square array whose n-th row lists all numbers having n as a substring, n >= 1; read by falling antidiagonals.

Original entry on oeis.org

1, 10, 2, 11, 12, 3, 12, 20, 13, 4, 13, 21, 23, 14, 5, 14, 22, 30, 24, 15, 6, 15, 23, 31, 34, 25, 16, 7, 16, 24, 32, 40, 35, 26, 17, 8, 17, 25, 33, 41, 45, 36, 27, 18, 9, 18, 26, 34, 42, 50, 46, 37, 28, 19, 10, 19, 27, 35, 43, 51, 56, 47, 38, 29, 100, 11

Offset: 1

M. F. Hasler, Oct 18 2017

			The array starts:
   [ 1  10  11  12  13  14  15  16  17  18  19  21  31 ...] = A011531
   [ 2  12  20  21  22  23  24  25  26  27  28  29  32 ...] = A011532
   [ 3  13  23  30  31  32  33  34  35  36  37  38  39 ...] = A011533
   [ 4  14  24  34  40  41  42  43  44  45  46  47  48 ...] = A011534
   [ 5  15  25  35  45  50  51  52  53  54  55  56  57 ...] = A011535
   [ 6  16  26  36  46  56  60  61  62  63  64  65  66 ...] = A011536
   [ 7  17  27  37  47  57  67  70  71  72  73  74  75 ...] = A011537
   [ 8  18  28  38  48  58  68  78  80  81  82  83  84 ...] = A011538
   [ 9  19  29  39  49  59  69  79  89  90  91  92  93 ...] = A011539
   [10 100 101 102 103 104 105 106 107 108 109 110 210 ...] = A293870
   [11 110 111 112 113 114 115 116 117 118 119 211 311 ...] = A293871
   [12 112 120 121 122 123 124 125 126 127 128 129 212 ...] = A293872
   [   ...             ...             ...             ...]

Rémy Sigrist, Table of n, a(n) for n = 1..5050
Rémy Sigrist, Perl program for A293869

Cf. A072484, A292690 (variant starting with row 0).
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539, A293870, A293871, A293872.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.
Cf. A179239, A261370.
Cf. A292451, A292731 (both partially coincide with row 11, but no inclusion relation holds).

Block[{d = 15, q, a, s}, a = Table[q = n-1; s = IntegerString[n]; Table[While[StringFreeQ[IntegerString[++q], s]]; q, d-n+1], {n, d}]; Table[a[[n, k-n+1]], {k, d}, {n, k}]] (* Paolo Xausa, Mar 01 2024 *)

has=(n,p,m=10^#Str(p))->until(p>n\=10,n%m==p&&return(1))
Mat(vectorv(12,n,a=[];for(k=n,oo,has(k,n)||next;a=concat(a,k);#a>12&&break);a))

Perl
```
See Links section.
```

T(n, k) = A072484(n, k) for any n > 0 and k = 1..n. - Rémy Sigrist, Jan 29 2021

A328447 Smallest representative of the class of numbers having the same digits as n up to permutation.

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, 12, 22, 23, 24, 25, 26, 27, 28, 29, 30, 13, 23, 33, 34, 35, 36, 37, 38, 39, 40, 14, 24, 34, 44, 45, 46, 47, 48, 49, 50, 15, 25, 35, 45, 55, 56, 57, 58, 59, 60, 16, 26, 36, 46, 56, 66, 67, 68, 69, 70, 17, 27, 37, 47, 57, 67

Offset: 0

M. F. Hasler, Oct 15 2019

Sort the digits in increasing order. If the list starts with a digit 0, move the smallest nonzero digit to the front.

Every term is in A179239. - David A. Corneth, Oct 17 2019

			a(201) = 102: largest digits go to the end, but the smallest nonzero digit must go first.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A179239, A004186 (largest representative of the class of n).

f:= proc(n) local L,i,t;
  L:= sort(convert(n,base,10));
  if L[1]=0 then
    t:= numboccur(0,L)+1;
    L:= [L[t],op(L[1..t-1]),op(L[t+1..-1])];
  fi;
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f(0):= 0:
map(f, [$0..100]);

Array[FromDigits@ If[First@ # == 0, Flatten@ MapAt[Reverse, TakeDrop[#, 2], 1], #] &@ Sort@ IntegerDigits[#] &, 67] (* Michael De Vlieger, Oct 17 2019 *)

A328447(n)={if(n=vecsort(digits(n)), n[1]|| for(k=2,#n,n[k]&&[n[1]=n[k],n[k]=0,break]));fromdigits(n)}

def A328447(n):
    if n == 0: return 0
    s = str(n)
    l, s = len(s), ''.join(sorted(s.replace('0','')))
    return int(s[0]+'0'*(l-len(s))+s[1:]) # Chai Wah Wu, Dec 06 2021

A039986 Primes such that every distinct permutation of digits is composite (including permutations with leading zeros).

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 151, 211, 223, 227, 229, 233, 257, 263, 269, 353, 383, 409, 431, 433, 443, 449, 487, 499, 523, 541, 557, 599, 661, 677, 773, 827, 829, 853, 859, 881, 883, 887, 929, 997, 1447, 1451, 1481, 2111

Offset: 1

David W. Wilson

At most one permutation of digits of A179239 can occur in this sequence. - David A. Corneth, Jun 28 2018
Is there a term with more than 4 distinct digits? - David A. Corneth, Jun 30 2018
Up through 9999991 (the largest 7-digit prime) there are no terms with more than 4 distinct digits. - Harvey P. Dale, Dec 12 2018
The sequence can be seen as a table with the n-digit terms in row n. Row lengths would then be (4, 13, 34, 45, 68, 67, 47, 36, 40, 46, 33, 45, 35, 38, 32, ...). In these rows there are (0, 0, 0, 6, 9, 3, 0, 1, 0, 0, ...) terms with >= 4 distinct digits: this seems to happen only for terms with 4, 5, 6 or 8 digits. I conjecture that there are no more than these 6 + 9 + 3 + 1 = 19 terms (2861, 4027, 4801, 5209, 5623, 5849, 24889, 26561, 40609, 40883, 66541, 66853, 85087, 85843, 86441, 288689, 442469, 558541, 55555429) with 4, and none with 5 or more distinct digits. - M. F. Hasler, Jul 01 2018
Prime repunits (A004022) are a subset of this sequence. As larger terms are seemingly all near-repdigit primes, it is possible to obtain very large terms. For example: (10^10002 - 1)/9 - 10^2872. - Hans Havermann, Jul 08 2018

Hans Havermann and M. F. Hasler, Table of n, a(n) for n = 1..1141 (Terms < 10^30; earlier terms from T. D. Noe and David A. Corneth.)
Hans Havermann, AEnlic primes.

Cf. A030291, A179239.
Cf. A225421 (only odd digits).
Cf. A244529 for another variant. - M. F. Hasler, Jun 28 2018

t = {}; Do[p=Prime[n]; If[Length[Select[Table[FromDigits[k], {k,Permutations[IntegerDigits[p]]}], PrimeQ]] == 1, AppendTo[t,p]], {n,330}]; t (* Jayanta Basu, May 07 2013 *)
Select[Prime[Range[400]],AllTrue[FromDigits/@Rest[ Permutations[ IntegerDigits[#]]], CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 22 2015 *)

is(n,d=digits(n))={isprime(n)&&!for(i=1,(#d)!, (n=vecextract(d,numtoperm(#d,i)))!=d&& isprime(fromdigits(n))&& return)} \\ Then: select(is,primes(500)) - M. F. Hasler, Jun 28 2018
is(n)={isprime(n)||return; my(d=vecsort(digits(n), (a, b)->if(a-b&& t=bittest(650, a)-bittest(650, b),t,a-b)), p=vector(#d,i,i), N(p,i=2)= while((t=p[i]-1)&& while((setsearch(Set(p[i+1..#p]),t)|| d[t]==d[p[i]])&& t--,); !t, i++>#p&& return); i<#p|| bittest(650, d[t])|| return; concat([setminus(Set(p[1..i]),[t]), t, p[i+1..#p]]), t); #d==1|| !until(!p=N(p),(n!=t=fromdigits(vecextract(d,p)))&& isprime(t)&& return)} \\ Produces only inequivalent permutations which can be prime. - M. F. Hasler, Jun 28 2018
A039986_row(n)={if(n>1, local(D=eval(Vec("0245681379")), u=vectorv(n, i, 10^(n-i)), nextperm()=for(i=2,n,(t=p[i]-1)&& while(setsearch(Set(p[i+1..n]),t)|| d[t]==d[p[i]], t--||break); t|| next; iM. F. Hasler, Jul 01 2018

from itertools import count, islice, combinations_with_replacement
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime
def A039986_gen(): # generator of terms
    for l in count(1):
        xlist = []
        for p in combinations_with_replacement('0123456789',l):
            flag = False
            for q in multiset_permutations(p):
                if isprime(m:=int(''.join(q))):
                    if flag or q[0]=='0':
                        flag = False
                        break
                    else:
                        flag = True
                        r = m
            if flag:
                xlist.append(r)
        yield from sorted(xlist)
A039986_list = list(islice(A039986_gen(),30)) # Chai Wah Wu, Dec 26 2023

Name clarified upon the suggestion of Robert Israel, Jun 30 2018

A035927 One less than number of n-multisets chosen from a 10-set.

Original entry on oeis.org

0, 9, 54, 219, 714, 2001, 5004, 11439, 24309, 48619, 92377, 167959, 293929, 497419, 817189, 1307503, 2042974, 3124549, 4686824, 6906899, 10015004, 14307149, 20160074, 28048799, 38567099, 52451255, 70607459, 94143279

Offset: 0

N. J. A. Sloane

Number of distinct n-digit numbers up to permutations of digits. - Michael Somos, Jul 11 2002

Equivalently, for n > 0, a(n) = number of n-digit decimal numbers d_1 d_2 ... d_n with d_1 > 0 and d_1 >= d_2 >= ... >= d_n >= 0.. - N. J. A. Sloane, Jul 13 2023

Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, ITEM 56, Cambridge, MA: Mass. Institute of Technology Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972, Item 56.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Equals A000582 - 1. Cf. A014553, A179239.

Maple
```
binomial(10+n-1,n)-1;
```

Table[Binomial[9 + n, n] - 1, {n, 0, 27}] (* Michael De Vlieger, Jul 14 2015 *)
CoefficientList[Series[1/(1-x)^10-1/(1-x),{x,0,30}],x] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,9,54,219,714,2001,5004,11439,24309,48619},30] (* Harvey P. Dale, Jul 11 2023 *)

PARI
```
a(n)=if(n<0,0,binomial(n+9,9)-1)
```

G.f.: 1/(1-x)^10-1/(1-x). - Michael Somos, Jul 11 2002

cp's OEIS Frontend

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

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A365375 Numbers being the smallest positive integer having its digits (Cf. A179239) from which two digits can be chosen, the difference being any value from 0 to 9.

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A047726 Number of different numbers that are formed by permuting digits of n.

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A101337 Sum of (each digit of n raised to the power (number of digits in n)).

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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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A099627 Triangle read by rows: T(n,k) = 2^n + 2^k - 1 with n >= k >= 0.

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