A028867 -id:A028867 - cp's OEIS frontend

A028864 Primes with digits in nondecreasing order.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 113, 127, 137, 139, 149, 157, 167, 179, 199, 223, 227, 229, 233, 239, 257, 269, 277, 337, 347, 349, 359, 367, 379, 389, 449, 457, 467, 479, 499, 557, 569, 577, 599, 677, 1117, 1123, 1129

Offset: 1

Patrick De Geest

Identical digits are acceptable, e.g., 1117 is in the sequence. - Harvey P. Dale, Aug 16 2011

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 1..1000 from T. D. Noe)

Cf. A009994, A052015, A028867, A052014, A028905.

[p:p in PrimesUpTo(1200)| Reverse(Intseq(p)) eq Sort(Intseq(p))]; // Marius A. Burtea, Nov 29 2019

daoQ[n_] := Count[Differences[IntegerDigits[n]], ?(# < 0 &)] == 0; Select[Prime[Range[200]], daoQ] (* _Harvey P. Dale, Aug 16 2011 *)
Select[Prime[Range[200]],Min[Differences[IntegerDigits[#]]]>-1&] (* Harvey P. Dale, Mar 02 2023 *)

select(n->n=digits(n); n==vecsort(n), primes(500)) \\ Charles R Greathouse IV, Mar 15 2014

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
def A028864_gen(): # generator of terms
    yield from (2,3,5,7)
    a, b = {'1':0,'2':1,'3':1,'4':2,'5':2,'6':2,'7':2,'8':3,'9':3}, (1,3,7,9)
    for l in count(1):
        for d in combinations_with_replacement('123456789',l):
            k = 10*int(''.join(d))
            for e in b[a[d[-1]]:]:
                if isprime(m:=k+e):
                    yield m
A028864_list = list(islice(A028864_gen(),30)) # Chai Wah Wu, Dec 25 2023

j=2; y=as.bigz(c()); while(j<1000) {
x=sort(as.numeric(strsplit(as.character(j),spl="")[[1]]),decr=F)
if(j==paste(x[x>0],collapse="")) y=c(y,j)
j=nextprime(j)
} //  Christian N. K. Anderson, Apr 04 2013

Trivially, a(n) >> exp(n^(1/10)). - Charles R Greathouse IV, Mar 15 2014

prime(n) = A028905(n) if prime(n) is in this sequence. - Alonso del Arte, Nov 25 2019

Definition corrected by Omar E. Pol, Mar 22 2012

A052015 Primes with distinct digits in ascending order.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579

Offset: 1

Patrick De Geest, Nov 15 1999

			Last term is a(100) = 23456789.

T. D. Noe, Table of n, a(n) for n = 1..100 (complete sequence)

Cf. A028864, A028867, A052014.

b:= proc(n) `if`(isprime(n), n, [][]), seq(
      b(parse(cat(n, j))), j=irem(n, 10)+1..9)
    end:
sort([seq(b(n), n=1..9)])[];  # Alois P. Heinz, Jun 16 2025

t={};Do[p=Prime[n];If[Select[Differences[IntegerDigits[p]],#<=0&]=={},AppendTo[t,p]],{n,380}];t (* Jayanta Basu, May 04 2013 *)
Select[Prime[Range[5000]],Min[Differences[IntegerDigits[#]]]>0&] (* Harvey P. Dale, Jun 20 2015 *)
Select[FromDigits@# &/@Subsets@Range@9,PrimeQ] (* Hans Rudolf Widmer, Apr 08 2023 *)

A052015=vecextract( vecsort( vector( 511,i,isprime( t=eval( concat( vecextract(Vec("123456789"),i ))))*t),NULL,8),"^1") /* for old PARI versions replace,NULL,8),"^1" by ),"-100.." */ \\ M. F. Hasler, Jan 27 2009

from sympy import isprime
from itertools import combinations
def agen(): # generator of terms
    for d in range(1, 9):
        for c in combinations("123456789", d):
            if isprime(t:=int("".join(c))):
                yield t
print(list(agen())) # Michael S. Branicky, Dec 13 2023

A052014 Primes with distinct digits in descending order.

Original entry on oeis.org

2, 3, 5, 7, 31, 41, 43, 53, 61, 71, 73, 83, 97, 421, 431, 521, 541, 631, 641, 643, 653, 743, 751, 761, 821, 853, 863, 941, 953, 971, 983, 5431, 6421, 6521, 7321, 7541, 7621, 7643, 8431, 8521, 8543, 8641, 8731, 8741, 8753, 8761, 9421, 9431, 9521, 9631, 9643

Offset: 1

Patrick De Geest, Nov 15 1999

Last term is a(87) = 98765431.

M. F. Hasler, Table of n, a(n) for n = 1..87. [The full list of terms.]

Cf. A000040, A028867, A028864, A052015.

b:= proc(n) `if`(isprime(n), n, [][]), seq(
      b(parse(cat(n, j))), j=1..irem(n, 10)-1)
    end:
sort([seq(b(n), n=1..9)])[];  # Alois P. Heinz, Jun 16 2025

Select[Prime[Range[1200]], Max[DigitCount[#]] == 1 && And@@Negative[ Differences[IntegerDigits[#]]]&] (* Harvey P. Dale, Oct 22 2011 *)
t={}; Do[p=Prime[n]; If[Select[Differences[IntegerDigits[p]], # >= 0&] == {}, AppendTo[t,p]], {n,1195}]; t (* Jayanta Basu, May 04 2013 *)

A052014=[]; for( i=1,1023, c=-1; isprime( t=sum( j=0,9, if(bittest(i,j),j*10^c++))) & A052014=concat(A052014,t)); A052014=vecsort(A052014)

A372034 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nonincreasing order.

Original entry on oeis.org

4, 8, 9, 22, 32, 33, 44, 55, 64, 77, 88, 93, 99, 422, 633, 775, 844, 933, 993, 4222, 4442, 6333, 6655, 6663, 7533, 7744, 7775, 8444, 8884, 9663, 9993, 44222, 66333, 88444, 99633, 99933, 99993, 933333, 966333, 996663, 999993, 4442222, 6663333, 7777775, 8884444, 9663333, 9666633, 9666663

Offset: 1

Scott R. Shannon, Apr 16 2024

Is it true that no terms end with 1? A separate search on those shows none with < 70 digits. Michael S. Branicky, Apr 23 2024

Testing all products of repunit primes (A004022, A004023), there are no terms ending in 1 less than 10^3000. - Michael S. Branicky, Apr 24 2024

			The initial terms and their factorizations are:
4 = [2, 2]
8 = [2, 2, 2]
9 = [3, 3]
22 = [2, 11]
32 = [2, 2, 2, 2, 2]
33 = [3, 11]
44 = [2, 2, 11]
55 = [5, 11]
64 = [2, 2, 2, 2, 2, 2]
77 = [7, 11]
88 = [2, 2, 2, 11]
93 = [3, 31]
99 = [3, 3, 11]
422 = [2, 211]
633 = [3, 211]
775 = [5, 5, 31]
844 = [2, 2, 211]
933 = [3, 311]
993 = [3, 331]
4222 = [2, 2111]
4442 = [2, 2221]
6333 = [3, 2111]
6655 = [5, 11, 11, 11]
6663 = [3, 2221]
7533 = [3, 3, 3, 3, 3, 31]
7744 = [2, 2, 2, 2, 2, 2, 11, 11]
...

Michael S. Branicky, Table of n, a(n) for n = 1..197 (all terms with <= 21 digits)

Cf. A004022, A004023, A009996, A028867, A372053, A372029, A027746, A372055.

from sympy import factorint, isprime
def ni(s): return sorted(s, reverse=True) == list(s)
def ok(n):
    if n < 4 or isprime(n): return False
    s, f = str(n), "".join(str(p)*e for p, e in factorint(n).items())
    return ni(s+f)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Apr 23 2024

# faster for initial segment of sequence
from sympy import factorint, isprime
from itertools import islice, combinations_with_replacement as mc
def ni(s): return s == "".join(sorted(s, reverse=True))
def bgen(d):
    yield from ("".join(m) for m in mc("987654321", d))
def agen(): # generator of terms
    for d in range(1, 70):
        out = set()
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if ni(s+"".join(str(p)*e for p, e in factorint(t).items())):
                out.add(t)
        yield from sorted(out)
print(list(islice(agen(), 50))) # Michael S. Branicky, Apr 23 2024

A211654 Primes that remain prime when their digits are sorted into nondecreasing order.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 47, 59, 67, 71, 73, 79, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 173, 179, 193, 197, 199, 223, 227, 229, 233, 239, 257, 269, 271, 277, 293, 307, 311, 317, 337, 347, 349, 359, 367, 373

Offset: 1

Francis J. McDonnell, Apr 17 2012

In sequence A004185 these are referred to as "sortable primes". Nontrivial terms (with digits not in nondecreasing order) are listed in A086042. - M. F. Hasler, Jul 30 2019.

			173 is prime and after the digits are sorted into nondecreasing order we obtain 137, which is prime.

Francis J. McDonnell, Table of n, a(n) for n = 1..10000
Francis J. McDonnell, Java Program

Cf. A028864, A028867, A211655.

Cf. A086042 (nontrivial solutions), A004185 (n with digits sorted).

[p:p in PrimesUpTo(400)| IsPrime(Seqint(Reverse(Sort(Intseq(p,10)))))]; // Marius A. Burtea, Jul 30 2019

Select[Prime[Range[200]], PrimeQ[FromDigits[Sort[IntegerDigits[#]]]] &] (* T. D. Noe, Apr 17 2012 *)

select( is_A211654(p)={isprime(fromdigits(vecsort(digits(p))))&&isprime(p)}, primes([1,999])) \\ M. F. Hasler, Jul 30 2019

A190220 Numbers all of whose divisors are numbers whose decimal digits are in nonincreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 31, 33, 40, 41, 43, 44, 53, 55, 61, 62, 63, 66, 71, 73, 77, 82, 83, 86, 88, 93, 97, 99, 110, 211, 220, 311, 331, 421, 422, 431, 433, 440, 443, 511, 521, 541, 622, 631, 633, 641, 643, 653, 661, 662, 733, 743, 751

Offset: 1

Jaroslav Krizek, May 06 2011

Subset of A009996. Superset of A028867, A190219 and A190217.

			Number 110 is in sequence because all divisors of 110 (1, 2, 5, 10, 11, 22, 55, 110) are numbers whose decimal digits are in nonincreasing order.

Nathaniel Johnston, Table of n, a(n) for n = 1..500

with(numtheory): A190220 := proc(n) option remember: local d, dd, i, j, k, m, poten: if(n=1)then return 1: fi: for k from procname(n-1)+1 do d:=divisors(k): poten:=1: for i from 1 to nops(d) do m:=-1: dd:=convert(d[i], base, 10): for j from 1 to nops(dd) do if(m<=dd[j])then m:=dd[j]: else poten:=0: break: fi: od: if(poten=0)then break:fi: od: if(poten=1)then return k: fi: od: end: seq(A190220(n), n=1..64); # Nathaniel Johnston, May 14 2011

A211655 Down-sortable primes: Primes that are also primes after digits are sorted into decreasing order.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 53, 61, 71, 73, 79, 83, 97, 113, 131, 149, 157, 163, 167, 179, 181, 191, 197, 199, 211, 241, 251, 281, 311, 313, 331, 337, 347, 359, 373, 389, 419, 421, 431, 433, 443, 461, 463, 491, 521, 541, 563, 571, 593, 613, 617, 631, 641, 643, 653

Offset: 1

Francis J. McDonnell, Apr 17 2012

All 1- and 2-digit reversible primes (A007500) are trivially in this sequence. No primes from A056709 are in this sequence. Clearly all absolute primes (A003459) are sortable primes but not all sortable primes are absolute primes. - Alonso del Arte, Oct 08 2013

			131 is prime and after sorting its digits into nonincreasing order we obtain 311, which is prime.
163 is in the sequence because its digits sorted in decreasing order give 631, which is prime. (Note that this is not a reversible prime, since 361 = 19^2.)

Francis J. McDonnell, Table of n, a(n) for n = 1..10000
Francis J. McDonnell, Java Program

Cf. A004186, A211654, A028864, A028867.

Select[Prime[Range[200]], PrimeQ[FromDigits[-Sort[-IntegerDigits[#]]]] &] (* T. D. Noe, Apr 17 2012 *)

A345326 Number of primes less than 10^n with digits in nonincreasing order.

Original entry on oeis.org

0, 4, 14, 49, 125, 296, 646, 1304, 2459, 4543, 7882, 13272, 21856, 34934, 53446, 82055, 121322, 175498, 251714, 354810, 488440, 676065, 914834, 1220629, 1627770, 2135954, 2759889, 3590609, 4602572, 5830588, 7386200, 9266652, 11469407, 14314939, 17658240

Offset: 0

Ilya Gutkovskiy, Jul 22 2021

Number of primes with at most n digits arranged in nonincreasing order.

Michael S. Branicky, Table of n, a(n) for n = 0..42
Index entries for sequences related to numbers of primes in various ranges

Cf. A000040, A006880, A028867, A345325.

Table[Length@Select[Prime@Range[PrimePi[10^n]],OrderedQ@Reverse@IntegerDigits@#&],{n,0,7}] (* Giorgos Kalogeropoulos, Jul 22 2021 *)

from sympy import isprime
from itertools import accumulate, combinations_with_replacement as mc
def numwithdigs(d):
    if d == 0: return 0
    nonincreasing = (int("".join(m)) for m in mc("987654321", d))
    return len(list(filter(isprime, nonincreasing)))
def aupto(nn): return list(accumulate(numwithdigs(d) for d in range(nn+1)))
print(aupto(14)) # Michael S. Branicky, Jul 22 2021

a(12)-a(34) from Michael S. Branicky, Jul 22 2021

A061245 Prime numbers with odd digits in descending order.

Original entry on oeis.org

3, 5, 7, 11, 31, 53, 71, 73, 97, 311, 331, 733, 751, 773, 911, 953, 971, 977, 991, 997, 3331, 5333, 5531, 7331, 7333, 7753, 9311, 9511, 9533, 9551, 9733, 9931, 9973, 33311, 33331, 55331, 55333, 55511, 73331, 75511, 75533, 75553, 77551, 77711

Offset: 1

Amarnath Murthy, Apr 23 2001

Cf. A061244, A028867, A052014.

More terms from Patrick De Geest, Jun 04 2001

A364831 Primes whose digits are prime and in nonincreasing order.

Original entry on oeis.org

2, 3, 5, 7, 53, 73, 733, 773, 5333, 7333, 7753, 55333, 75533, 75553, 77773, 733333, 755333, 775553, 7553333, 7555333, 7775533, 7777753, 55555333, 55555553, 77755553, 555553333, 755555533, 773333333, 777555553, 777773333, 777775333, 777775553, 777777773

Offset: 1

James C. McMahon, Aug 09 2023

Intersection of A028867 and A019546.

The subsequence for primes whose digits are prime and in strictly decreasing order has just six terms: 2 3 5 7 53 73 (see A177061).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A009996, A019546, A177061, A028867.

Select[Prime[Range[3100000]], AllTrue[d = IntegerDigits[#], PrimeQ] && GreaterEqual @@ d &]

from itertools import count, islice, chain, combinations_with_replacement
from sympy import isprime
def A364831_gen(): # generator of terms
    yield 2
    yield from chain.from_iterable((sorted(s for d in combinations_with_replacement('753',l) if isprime(s:=int(''.join(d)))) for l in count(1)))
A364831_list = list(islice(A364831_gen(),30)) # Chai Wah Wu, Sep 10 2023

cp's OEIS Frontend

A028864 Primes with digits in nondecreasing order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

R

Formula

Extensions

A052015 Primes with distinct digits in ascending order.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A052014 Primes with distinct digits in descending order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A372034 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nonincreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Python

A211654 Primes that remain prime when their digits are sorted into nondecreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A190220 Numbers all of whose divisors are numbers whose decimal digits are in nonincreasing order.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

A211655 Down-sortable primes: Primes that are also primes after digits are sorted into decreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs