A052015 -id:A052015 - 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

A028867 Primes with digits in nonincreasing order.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 41, 43, 53, 61, 71, 73, 83, 97, 211, 311, 331, 421, 431, 433, 443, 521, 541, 631, 641, 643, 653, 661, 733, 743, 751, 761, 773, 811, 821, 853, 863, 877, 881, 883, 887, 911, 941, 953, 971, 977, 983, 991, 997, 2111, 2221, 3221

Offset: 1

Patrick De Geest

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

Cf. A052014, A028864, A052015.

t={};Do[p=Prime[n];If[Select[Differences[IntegerDigits[p]],#>0&]=={},AppendTo[t,p]],{n,460}];t (* Jayanta Basu, May 04 2013 *)
Select[Prime[Range[600]],Max[Differences[IntegerDigits[#]]]<1&] (* Harvey P. Dale, Oct 30 2013 *)

is(n)=my(d=digits(n)); for(i=2,#d,if(d[i]>d[i-1],return(0))); isprime(n) \\ Charles R Greathouse IV, Aug 18 2017

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
def A028867_gen(): # generator of terms
    yield from (2,3,5,7)
    a, b = {'1':1,'2':1,'3':2,'4':2,'5':2,'6':2,'7':3,'8':3,'9':4}, (1,3,7,9)
    for l in count(1):
        mlist = []
        for d in combinations_with_replacement('987654321',l):
            k = 10*int(''.join(d))
            for e in b[:a[d[-1]]]:
                if isprime(m:=k+e):
                    mlist.append(m)
        yield from sorted(mlist)
A028867_list = list(islice(A028867_gen(),30)) # Chai Wah Wu, Dec 25 2023

Definition corrected by Omar E. Pol, Mar 22 2012

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)

A071363 Largest n-digit prime with strictly increasing digits.

Original entry on oeis.org

7, 89, 569, 5689, 34679, 345689, 1456789, 23456789

Offset: 1

Rick L. Shepherd, May 21 2002

Notice the terms with consecutive digits; search for 23456789 to find several related sequences including A006055, A052017 and A052077.

			a(1) = A052015(4), a(2) = A052015(15), a(3) = A052015(35), a(4) = A052015(61), ... In short, a(n) = A052015(b(n)) with b = (4, 15, 35, 61, 81, 94, 98, 100). - _M. F. Hasler_, May 03 2017

Cf. A071362, A071360, A071361, A007810.

Subsequence of A052015.

A071363(n,u=vectorv(n,i,10^(n-i)))={forvec(d=vector(n,i,[1,9]),isprime(d*u)&&n=d*u,2);n} \\ M. F. Hasler, May 03 2017

A155763 Indices of primes with digits in strictly increasing order.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 15, 17, 19, 22, 24, 31, 33, 34, 35, 37, 39, 41, 52, 55, 57, 69, 70, 72, 73, 75, 77, 88, 91, 92, 104, 203, 204, 205, 207, 209, 219, 232, 237, 247, 249, 278, 348, 350, 355, 364, 365, 376, 391, 405, 483, 486, 487, 619, 633, 644, 749, 1475

Offset: 1

Zak Seidov, Jan 27 2009

Indices of primes in A052015.

There are exactly 100 terms: 1,2,3,4,6,7,8,9,10,12,15,17,19,22,24,31,33,34,35,37,39,41,52,55,57,69,70,72,73,75,77,88,91,92,104,203,204,205,207,209,219,232,237,247,249,278,348,350,355,364,365,376,391,405,483,486,487,619,633,644,749,1475,1478,1487,1489,1501,1504,1515,1595,1597,1605,1615,1631,1830,2611,2622,2637,2646,2829,3694,3704,11602,11603,11699,11706,11803,12560,13479,20797,20884,20891,21698,29629,29630,95375,95443,96140,111119,809912,1475171.

Harvey P. Dale, Table of n, a(n) for n = 1..100

A052015

Select[Range[1500],Min[Differences[IntegerDigits[Prime[#]]]]>0&] (* Harvey P. Dale, Feb 27 2017 *)

A190218 Numbers all of whose divisors are numbers whose decimal digits are in strictly increasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 125, 127, 134, 135, 136, 137, 138, 139, 145, 149, 157, 158, 167, 169, 178, 179, 235

Offset: 1

Jaroslav Krizek, May 06 2011

Sequence is finite. Last term a(163) = 23456789.

Subset of A009993. Superset of A052015.

			Number 135 is in sequence because all divisors of 135 (1, 3, 5, 9, 15, 27, 45, 135) are numbers whose decimal digits are in strictly increasing order.

Nathaniel Johnston and Jaroslav Krizek, Table of n, a(n) for n = 1..163 (complete list)

with(numtheory): A190218 := 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(A190218(n), n=1..62); # Nathaniel Johnston, May 06 2011

Select[Range[250], And@@Positive[Flatten[Differences/@(IntegerDigits/@Divisors[#])]]&] (* Harvey P. Dale, Mar 24 2012 *)

A061244 Prime numbers with odd digits in ascending order.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 37, 59, 79, 113, 137, 139, 157, 179, 199, 337, 359, 379, 557, 577, 599, 1117, 1399, 1559, 1579, 1777, 1999, 3359, 3557, 3559, 3779, 5557, 5779, 11113, 11117, 11119, 11159, 11177, 11399, 11579, 11777, 11779, 13337, 13339

Offset: 1

Amarnath Murthy, Apr 23 2001

			37 is a term since 3 and 7 are odd digits and 3 < 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A061245, A028864, A052015.

A061244:= {}:
for d from 1 to 20 do
  for n1 from 0 to d do
    x1:= (10^n1-1)/9;
    for n3 from 0 to d-n1 do
      x3:= x1*10^n3 + (10^n3-1)/9*3;
      for n5 from 0 to d-n1-n3 do
        x5:= x3*10^n5 + (10^n5-1)/9*5;
        for n7 from 0 to d-n1-n3-n5 do
          x7:= x5*10^n7 + (10^n7-1)/9*7;
          n9:= d-n1-n3-n5-n7;
          x:= x7*10^n9 + (10^n9-1);
          if isprime(x) then
            A061244:= A061244 union {x};
          fi
od od od od od:
A061244;
# Robert Israel, Apr 20 2014

More terms from Patrick De Geest, Jun 04 2001

A211771 Nonprime numbers with distinct digits in ascending order.

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 14, 15, 16, 18, 24, 25, 26, 27, 28, 34, 35, 36, 38, 39, 45, 46, 48, 49, 56, 57, 58, 68, 69, 78, 123, 124, 125, 126, 128, 129, 134, 135, 136, 138, 145, 146, 147, 148, 156, 158, 159, 168, 169, 178, 189, 234, 235, 236, 237, 238, 245, 246, 247

Offset: 1

Jaroslav Krizek, May 07 2012

Sequence is finite with 411 terms, last term is a(411) = 123456789.

Complement of A052015 with respect to A009993. Supersequence of A211772.

Jaroslav Krizek, Table of n, a(n) for n = 1..411 (complete list)

Cf. A052015 (primes with distinct digits in ascending order), A009993 (numbers with distinct digits in ascending order), A211772 (nonprime numbers all of whose divisors are numbers whose decimal digits are in ascending order).

nnddQ[n_]:=!PrimeQ[n]&&Max[DigitCount[n]]<2&&Min[Differences[ IntegerDigits[ n]]]>0; Select[Range[300],nnddQ] (* Harvey P. Dale, Sep 27 2020 *)

A178788(a(n)) = 1.

A211772 Nonprime numbers all of whose divisors are numbers whose decimal digits are in ascending order.

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 14, 15, 16, 18, 24, 25, 26, 27, 28, 34, 35, 36, 38, 39, 45, 46, 48, 49, 56, 57, 58, 68, 69, 78, 125, 134, 135, 136, 138, 145, 158, 169, 178, 235, 237, 245, 247, 259, 267, 268, 278, 289, 356, 358, 469, 478, 578, 1345, 1357, 1369, 2479, 2569

Offset: 1

Jaroslav Krizek, May 07 2012

Sequence is finite with 63 terms, last term is a(63) = 134689.

Complement of A052015 with respect to A190218. Subsequence of A211771.

			Divisors of 24589: 1, 67, 367, 24589 (all divisors with digits in ascending order).

Jaroslav Krizek, Table of n, a(n) for n = 1..63 (complete list).

Cf. A052015 (primes with distinct digits in ascending order), A190218 (numbers all of whose divisors are numbers whose decimal digits are in ascending order), A211771 (nonprime numbers with distinct digits in ascending order).

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

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A028864 Primes with digits in nondecreasing order.

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A028867 Primes with digits in nonincreasing order.

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A052014 Primes with distinct digits in descending order.

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A071363 Largest n-digit prime with strictly increasing digits.

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A155763 Indices of primes with digits in strictly increasing order.

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A190218 Numbers all of whose divisors are numbers whose decimal digits are in strictly increasing order.

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A061244 Prime numbers with odd digits in ascending order.

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A211771 Nonprime numbers with distinct digits in ascending order.

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