A028864 -id:A028864 - cp's OEIS frontend

A190221 Numbers all of whose divisors are numbers whose decimal digits are in nondecreasing order.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 125

Offset: 1

Jaroslav Krizek, May 06 2011

Subset of A009994. Superset of A028864, A190218 and A190217.

			Number 112 is in sequence because all divisors of 112 (1, 2, 4, 7, 8, 14, 16, 28, 56, 112) are numbers whose decimal digits are in nondecreasing order.

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

with(numtheory): A190221 := 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:=10: 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(A190221(n), n=1..64); # Nathaniel Johnston, May 06 2011

ndoQ[n_]:=Min[Differences[IntegerDigits[n]]]>=0; Select[Range[ 200],AllTrue[ Divisors[#],ndoQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 02 2021 *)

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 *)

A345325 Number of primes less than 10^n with digits in nondecreasing order.

Original entry on oeis.org

0, 4, 16, 50, 132, 315, 689, 1413, 2636, 4967, 8563, 14481, 23593, 37127, 56809, 86779, 127096, 184517, 264288, 368794, 510442, 707483, 948307, 1268871, 1689642, 2204795, 2866855, 3729223, 4738019, 6013021, 7619227, 9510372, 11832748, 14770667, 18067652

Offset: 0

Ilya Gutkovskiy, Jul 22 2021

Number of primes with at most n digits arranged in nondecreasing 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, A028864, A345326.

Table[Length@Select[Prime@Range[PrimePi[10^n]],OrderedQ@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("123456789", 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(11)-a(34) from Michael S. Branicky, Jul 22 2021

A343834 Primes with digits in nondecreasing order, only primes, and with sum of digits also a prime.

Original entry on oeis.org

2, 3, 5, 7, 23, 223, 227, 337, 557, 577, 2333, 2357, 2377, 2557, 2777, 33377, 222337, 222557, 233357, 233777, 235577, 2222333, 2233337, 2235557, 3337777, 3355777, 5555777, 22222223, 22233577, 23333357, 23377777, 25577777, 222222227, 222222557, 222222577

Offset: 1

Mikk Heidemaa, May 01 2021

Intersection of A028864 and A062088.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A019546, A028864, A046704, A062088.

a[p_] := With[{dg = IntegerDigits@p}, PrimeQ@p && OrderedQ@dg && AllTrue[dg, PrimeQ] && PrimeQ@ Total@dg]; Cases[ Range[3*10^7], _?(a@# &)] (* or *)
upToDigitLen[k_] := Cases[ FromDigits@# & /@ Select[ Flatten[ Table[ Tuples[{2, 3, 5, 7}, {i}], {i, k}], 1], OrderedQ[#] &], _?(PrimeQ@# && PrimeQ@ Total@ IntegerDigits@# &)]; upToDigitLen[10]

from sympy import isprime
from sympy.utilities.iterables import multiset_combinations
def aupton(terms):
  n, digits, alst = 0, 1, []
  while len(alst) < terms:
    mcstr = "".join(d*digits for d in "2357")
    for mc in multiset_combinations(mcstr, digits):
      sd = sum(int(d) for d in mc)
      if not isprime(sd): continue
      t = int("".join(mc))
      if isprime(t): alst.append(t)
      if len(alst) == terms: break
    else: digits += 1
  return alst
print(aupton(35)) # Michael S. Branicky, May 01 2021

a(33) and beyond from Michael S. Branicky, May 01 2021

A364458 Prime numbers that are not repdigits with digits in nondecreasing order with the property that any nontrivial permutation of the digits gives a composite number.

Original entry on oeis.org

19, 23, 29, 47, 59, 67, 89, 223, 227, 229, 233, 257, 269, 449, 499, 557, 599, 677, 1447, 2267, 2447, 4447, 5557, 8999, 11119, 15559, 22229, 22669, 23333, 24889, 44449, 48889, 55589, 55889, 59999, 79999, 222269, 444449, 455557, 555557, 555589, 666667, 4444469, 4555559

Offset: 1

Jean-Marc Rebert, Dec 23 2023

The least terms with respectively 2, 3, 4 distinct digits are 19, 257, 24889.

			19 is a term, because the digits of 19 are in nondecreasing order and 91 is the unique number != 19 given by a permutation of 19 and 91 = 7 * 13 is composite and the digits of 91 are not in nondecreasing order.

Michael S. Branicky, Table of n, a(n) for n = 1..108 (all terms with <= 52 digits)
David A. Corneth, PARI program

Cf. A028864, A039986.

is(k)=my(u=digits(k),n=#u);if(#vecsort(u,,8)==1||u!=vecsort(u)||!isprime(k),return(0));forperm(n,p,my(vp=Vec(p),v=[]);for(i=1,n,v=concat(v,u[vp[i]]));q=fromdigits(v);if(k!=q&&isprime(q),return(0)));1

PARI
```
\\ See PARI link
```

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations as mp
from itertools import count, islice, combinations_with_replacement as mc
def bgen(d): yield from ("".join(m) for m in mc("123456789", d))
def agen(): yield from (t for d in count(1) for k in bgen(d) if len(set(k))!=1 and isprime(t:=int(k)) if not any((j:="".join(m))!=k and isprime(int(j)) for m in mp(k)))
print(list(islice(agen(), 44))) # Michael S. Branicky, Dec 23 2023

A381158 Prime numbers where digit values decrease while alternating parity.

Original entry on oeis.org

2, 3, 5, 7, 41, 43, 61, 83, 521, 541, 743, 761, 941, 983, 6521, 8521, 8543, 8741, 8761, 76541, 76543, 94321, 98321, 98543

Offset: 1

James S. DeArmon, Feb 15 2025

			41 is a term, since the digits decrease in value and alternate even and odd.

Cf. A052014, A000040, A028867, A028864, A052015, A382027.

b:= proc(n) `if`(isprime(n), n, [][]), seq(b(10*n+j), j=irem(n, 10)-1..1, -2) end:
{seq(b(n), n=1..9)}[];  # Alois P. Heinz, Mar 12 2025

ad=Select[Prime[Range[10^6]],Reverse[Union[IntegerDigits[#]]]==IntegerDigits[#]&];fQ[n_] := Block[{m = Mod[ IntegerDigits@ n, 2]}, m == Split[m, UnsameQ][[1]]];Select[ad,fQ] (* James C. McMahon, Mar 20 2025 *)

from sympy import isprime
from itertools import combinations
def ap(s): return all((s[i] in "13579") == (s[i+1] in "02468") for i in range(len(s)-1))
def afull(): return sorted(t for d in range(1, 10) for c in combinations("987654321", d) if ap(c) and isprime(t:=int("".join(c))))
print(afull()) # Michael S. Branicky, Mar 10 2025

A155775 Indices j in A000040 such that j is an odd composite and the distinct digits of the prime A000040(j) are in increasing order.

Original entry on oeis.org

9, 15, 33, 35, 39, 55, 57, 69, 75, 77, 91, 203, 205, 207, 209, 219, 237, 247, 249, 355, 365, 391, 405, 483, 633, 749, 1475, 1501, 1515, 1595, 1605, 1615, 1631, 2611, 2637, 2829, 11603, 11803, 13479, 20797, 20891, 95375

Offset: 1

Juri-Stepan Gerasimov, Jan 27 2009

These are the odd composites n (A071904) such that the digits of prime(n) are strictly increasing. The latter defines a subsequence of A028864, which lists primes with nondecreasing digits. - R. J. Mathar, Jan 30 2009

			If odd number=n=9 and prime(9)=23, then 2<3 and 9=a(1). If odd number=n=15 and prime(15)=47, then 4<7 and 15=a(2). If odd number=n=33 and prime(33)=137, then 1<3<7 and 33=a(3), etc.

Cf. A000040, A002808, A155079.

Corrected and extended by R. J. Mathar, Jan 30 2009

Better definition from Omar E. Pol, Jan 31 2009

A296520 Multidigit primes in which the largest digit appears only once.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 211, 223, 227, 229, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Zak Seidov, Dec 14 2017

			In 13 the last digit is the largest, in 131 the middle digit and in 41 the first digit.

A156666 is a subsequence (except that the single-digit primes are included).

Cf. A028864, A052015, A106118, A106119.

Reap[Do[p=Prime[k];id=IntegerDigits[p];If[Count[id,Max[id]]==1,Sow[p]],{k,6,100}]][[2,1]]
(* Second program: *)
Select[Prime@ Range[5, 60], Last@ Select[DigitCount[#], # > 0 &] == 1 &] (* Michael De Vlieger, Dec 14 2017 *)

A352721 Perfect cubes whose decimal digits appear in nonincreasing order.

Original entry on oeis.org

0, 1, 8, 64, 1000, 8000, 64000, 1000000, 8000000, 64000000, 1000000000, 8000000000, 64000000000, 1000000000000, 8000000000000, 64000000000000, 1000000000000000, 8000000000000000, 64000000000000000, 1000000000000000000, 8000000000000000000, 64000000000000000000

Offset: 1

Antonio Roldán, Mar 30 2022

			64 is in the sequence because it is a perfect cube (64 = 4^3) whose digits appear in nonincreasing order.

Cf. A004647, A028820, A028864, A234848, A273045.

Intersection of A000578 and A009996.

Select[Range[0, 4*10^6]^3, Max@ Differences[IntegerDigits[#]] <= 0 &] (* Amiram Eldar, Mar 30 2022 *)

ok(n) = digits(n) == vecsort(digits(n),,4) && ispower(n,3)

a(n) = A004647(n-1)^3.

cp's OEIS Frontend

A190221 Numbers all of whose divisors are numbers whose decimal digits are in nondecreasing order.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A345325 Number of primes less than 10^n with digits in nondecreasing order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A343834 Primes with digits in nondecreasing order, only primes, and with sum of digits also a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A364458 Prime numbers that are not repdigits with digits in nondecreasing order with the property that any nontrivial permutation of the digits gives a composite number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

A381158 Prime numbers where digit values decrease while alternating parity.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

A155775 Indices j in A000040 such that j is an odd composite and the distinct digits of the prime A000040(j) are in increasing order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A296520 Multidigit primes in which the largest digit appears only once.

Views

Author

Keywords

Examples

Crossrefs

Programs