A009996 -id:A009996 - cp's OEIS frontend

A261020 Numbers k such that the set of the decimal digits is a subgroup of the multiplicative group (Z/mZ)* where m is the sum of the decimal digits of k.

11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 111, 124, 139, 142, 193, 214, 241, 319, 391, 412, 421, 913, 931, 1111, 1115, 1133, 1151, 1155, 1177, 1199, 1248, 1284, 1313, 1331, 1379, 1397, 1428, 1482, 1511, 1515, 1551, 1717, 1739, 1771, 1793

Offset: 1

Michel Lagneau, Aug 07 2015

(Z/mZ)* is the multiplicative group of units of Z/mZ.
Let d(1)d(2)...d(q) be the q decimal digits of a number k. The principle of the algorithm is to compute all the products d(i)*d(j) (mod m) for 1 <= i,j <= q, and also the multiplicative inverse of each element such that if x is in the group, then there exists x' in the group where x*x' = 1.
The sequence is infinite because the numbers 11, 111, 1111, ... are in the sequence and generate the trivial subgroup {1}.
Only zerofree elements of A009996 have to be checked. Terms that match the criterion and permutations of their digits form all terms of this sequence due to commutativity of multiplication. - David A. Corneth, Aug 08 2015
To reduce cases, only check terms from A009995 (containing a 1 but no 0) for values m from digsum(term) to 81. - David A. Corneth, Aug 13 2015
Each decimal digit must be relatively prime to the decimal digit sum. - Tom Edgar, Aug 17 2015

			139 is a term because 1+3+9 = 13 and the elements {1, 3, 9} form a subgroup of the multiplicative group (Z/13Z)* with 12 elements. Each element is invertible: 1*1 == 1 (mod 13), 3*9 == 1 (mod 13) and 9*3 == 1 (mod 13). The other numbers of the sequence having the same property with (Z/13Z)* are 139, 193, 319, 391, 913, and 931.
1248 is in the sequence because 1+2+4+8 = 15 and the elements {1, 2, 4, 8} form a subgroup of the multiplicative group (Z/15Z)* with 8 elements: {1,2,4,7,8,11,13,14}.

Michel Lagneau and David A. Corneth, Table of n, a(n) for n = 1..10083 (all elements < 10^14, first 268 terms from Michel Lagneau)
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group

Cf. A011531, A261021, A261322.

nn:=2000:
for n from 1 to nn do:
x:=convert(n,base,10):nn0:=length(n):
lst1:={op(x),x[nn0]}:n0:=nops(lst1):
s:=sum('x[i]', 'i'=1..nn0):lst:={}:
   if  lst1[1]=1 then
    for j from 1 to n0 do:
     for l from j to n0 do:
      p:=irem(lst1[j]*lst1[l],s):lst:=lst union {p}:
     od:
    od:
    if lst=lst1
     then
       n3:=nops(lst1):lst2:={}:
        for c from 1 to n3 do:
          for d from 1 to n3 do:
           if irem(lst1[c]*lst1[d], s)=1
            then
            lst2:=lst2 union {lst1[c]}:
            else
           fi:
          od:
         od:
           if lst2=lst
           then
           printf(`%d, `, n):
           else
           fi:
          fi:
         fi:
     od:

def is_group(n):
    DD=n.digits()
    digsum=sum(DD)
    D=Set(DD)
    if not(1 in D) or 0 in D:
        return false
    for x in D:
        for y in D:
            if not(gcd(y,digsum)==1):
                return false
            if not((x*inverse_mod(y,digsum))%digsum in D):
                return false
    return true
[n for n in [1..2000] if is_group(n)] # Tom Edgar, Aug 17 2015

A261021 a(1)=0; for n > 1, a(n) is the number k such that the set of the decimal digits is an additive group Z/mZ where m is the sum of the decimal digits.

Original entry on oeis.org

0, 101, 102, 110, 120, 201, 202, 204, 210, 220, 240, 303, 306, 330, 360, 402, 404, 408, 420, 440, 480, 505, 550, 603, 606, 630, 660, 707, 770, 804, 808, 840, 880, 909, 990, 1001, 1002, 1010, 1020, 1100, 1200, 2001, 2002, 2004, 2010, 2020, 2040, 2100, 2200, 2400

Offset: 1

Michel Lagneau, Aug 07 2015

By convention, a(1)=0 because the trivial additive group is usually denoted by 0 where 0 is the identity element.
Let d(1)d(2)..d(q) be the q decimal digits of a number k. The principle of the algorithm is to compute all the sums (d(i)+ d(j))/mZ for 1 <= i,j <= q, and also the additive inverse of each element such that if x is in the group, then there exists x' in the group where x+x' = 0.
The sequence is infinite because the numbers 101, 1001, 10001, ... are in the sequence and generate the group {0,1}.
Only terms of A009996 containing at least one 0 have to be checked. Terms that match the criterion and numbers containing at least one 0 formed by permutations of their digits form all terms of this sequence due to commutativity of addition. - David A. Corneth, Aug 13 2015

			408 is in the sequence because 4+0+8 = 12 and the elements {0, 4, 8} is an additive group, subgroup of (Z/12Z,+) with 6 elements {0, 2, 4, 6, 8, 10}. Each element has an inverse: 2+10 == 0 (mod 12), 4+8 == 0 (mod 12), 6+6 == 0 (mod 12), 8+4 == 0 (mod 12) and 10+2 == 0 (mod 12).
The subsequence having the same property with Z/12Z is {408, 480, 606, 660, 804, 840, 4008, 4080, 4800, 6006, 6060, 6600, 8004, 8040, 8400, 40008, 40080, 40800, 48000, 60006, 60060, 60600, 66000, 80004, 80040, 80400, 84000, ...}.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 281 terms from Michel Lagneau)
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group

Cf. A009996, A261020.

nn:=3000:
for n from 1 to nn do:
x:=convert(n,base,10):nn0:=length(n):
lst1:={op(x),x[nn0]}:n0:=nops(lst1):
s:=sum('x[i]', 'i'=1..nn0):lst:={}:
   if  lst1[1]=0 then
    for j from 1 to n0 do:
     for l from j to n0 do:
      p:=irem(lst1[j]+lst1[l],s):lst:=lst union {p}:
     od:
    od:
    if lst=lst1
     then
       n3:=nops(lst1):lst2:={}:
        for c from 1 to n3 do:
          for d from 1 to n3 do:
           if irem(lst1[c]+lst1[d], s)=0
            then
            lst2:=lst2 union {lst1[c]}:
            else
           fi:
          od:
         od:
           if lst2=lst
           then
           printf(`%d, `, n):
           else
           fi:
          fi:
         fi:
     od:

is(n) = {my(d = digits(n),s = Set(digits(n))); if(n==0,return(1));
if(#s==2 || #s==3,return(s[1]==0 && (s[#s] / s[2] == 2^(#s-2)) && hammingweight(d)==2),return(0))}
\\a(n) works for n > 1.
a(n) = {my(qd = ((-1 + sqrt(1 + 8*(n + 15+1/2) / 17)) / 2)\1 + 2, v = vector(qd),i=1,h=2); n -= (binomial(qd-1,2)*17 -16); while(n-(qd-1)*h>0,
n-=(qd-1)*h;i++; h=1 + (i%2 == 0) + (i < 5)); n--; v[1]=i;
v[qd-n\h] = i*2^(n%h-(i%2==0)); sum(i=1,#v,10^(#v-i)*v[i])} \\ David A. Corneth, Aug 13 2015

For d >= 3, there are (d - 1) * 17 terms having d digits. - David A. Corneth, Aug 13 2015

A273046 Fibonacci numbers with digits in nonincreasing order.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 21, 55, 610, 987

Offset: 1

Omar E. Pol, May 13 2016

Presumably there are no more terms in this sequence. - Charles R Greathouse IV, May 17 2016

Cf. A000045, A009996, A272918, A273045.

Select[Fibonacci@ Range[0, 10^4], Reverse@ Sort@ # == # &@ IntegerDigits@ # &] (* Michael De Vlieger, May 13 2016 *)

fibmod(n,m)=lift(((Mod([1,1;1,0],m))^n)[1,2])
isA009996(n)=my(d=digits(n)); vecsort(d,,4)==d
B1=10^9;B2=10^57;
for(n=1,1e9, if(isA009996(fibmod(n,B1)) && isA009996(fibmod(n,B2)) && isA009996(F=fibonacci(n)), print1(F", "))) \\ Charles R Greathouse IV, May 17 2016

A355063 Perfect powers whose digits are in nonincreasing order and do not include 0.

Original entry on oeis.org

1, 4, 8, 9, 32, 64, 81, 441, 841, 961, 7744, 7776, 8874441, 9853321, 999887641

Offset: 1

Jon E. Schoenfield, Jun 16 2022

a(16) > 10^45 if it exists. - Michael S. Branicky, Jun 19 2022

Cf. A001597, A009996, A028822, A062826.

perfectPowerQ[n_] := n==1||GCD @@ FactorInteger[n][[All, 2]] > 1; (* A001597 *) Select[Range[10^5], perfectPowerQ[#] && Max[Differences[d=IntegerDigits[#]]]<1 && Count[d,0]==0&] (* Stefano Spezia, Jul 01 2025 *)

isok(m) = if (ispower(m), my(d=digits(m)); vecmin(d) && (d == vecsort(d,,4))); \\ Michel Marcus, Jun 17 2022

from sympy import perfect_power as pp
from itertools import count, islice, combinations_with_replacement as mc
def agen():
    yield 1
    for d in count(1):
        nd = (int("".join(m)) for m in mc("987654321", d))
        yield from sorted(filter(pp, nd))
print(list(islice(agen(), 14))) # Michael S. Branicky, Jun 16 2022

A378808 Numbers with monotonically decreasing digits, decreasing by only 0 or 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 32, 33, 43, 44, 54, 55, 65, 66, 76, 77, 87, 88, 98, 99, 100, 110, 111, 210, 211, 221, 222, 321, 322, 332, 333, 432, 433, 443, 444, 543, 544, 554, 555, 654, 655, 665, 666, 765, 766, 776, 777, 876, 877, 887, 888, 987, 988, 998, 999

Offset: 1

Randy L. Ekl, Dec 07 2024

			32 is a term since it has monotonically decreasing digits whose difference is at most 1.
33 is a term since it also has monotonically decreasing digits whose difference is at most 1.

Cf. A009996, A064222, A378775.

Select[Range[999],SubsetQ[{0,1},-Differences[IntegerDigits[#]]] &] (* Stefano Spezia, Dec 08 2024 *)

from itertools import count, islice
def bgen(last, d):
    if d == 0: yield tuple(); return
    t = (1, 9) if last == None else (max(0, last-1), last)
    for i in range(t[0], t[1]+1): yield from ((i,)+r for r in bgen(i, d-1))
def agen(): # generator of terms
    yield from (int("".join(map(str, i))) for d in count(1) for i in bgen(None, d))
print(list(islice(agen(), 62))) # Michael S. Branicky, Dec 08 2024

A265324 Numbers with nonincreasing digits such that every cyclic shift is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 73, 97, 311, 733, 971, 991, 9311, 999331, 1111111111111111111, 11111111111111111111111

Offset: 1

Chai Wah Wu, Dec 07 2015

a(18) is too big to display, see the b-file.

a(n) is the intersection of the sequences A068652 and A009996.

Chai Wah Wu, Table of n, a(n) for n = 1..18

Cf. A068652, A009996, A263499.

A344887 a(n) is the least base k >= 2 that the base-k digits of n are nonincreasing.

Original entry on oeis.org

2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 4, 5, 2, 3, 2, 2, 2, 5, 3, 6, 4, 3, 3, 5, 2, 3, 3, 3, 2, 7, 2, 2, 2, 6, 6, 6, 3, 4, 7, 3, 3, 4, 4, 6, 7, 7, 7, 7, 2, 7, 5, 8, 4, 4, 3, 5, 2, 4, 4, 8, 2, 4, 2, 2, 2, 9, 3, 3, 9, 9, 9, 10, 3, 8, 9, 3, 3, 9, 3, 3, 3, 3, 10, 10, 4, 4

Offset: 0

Rémy Sigrist, Jun 01 2021

			For n = 258:
- we have:
     b  258 in base b  Nonincreasing?
     -  -------------  --------------
     2      100000010  No
     3         100120  No
     4          10002  No
     5           2013  No
     6           1110  Yes
- so a(258) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A000196, A009996, A023758, A273040.

Table[k=1;While[AnyTrue[Differences@IntegerDigits[n,++k],#>0&]];k,{n,0,100}] (* Giorgos Kalogeropoulos, Jun 02 2021 *)

a(n) = { for (b=2, oo, my (d=digits(n, b)); if (d==vecsort(d,,4), return (b))) }

# with library / without (faster for large n)
from sympy.ntheory import digits
def is_nondec(n, b): d = digits(n, b)[1:]; return d == sorted(d)[::-1]
def is_nondec(n, b):
  if n < b: return True
  n, r = divmod(n, b)
  while n >= b:
    (n, r), lastd = divmod(n, b), r
    if r < lastd: return False
  return n >= r
def a(n):
  for b in range(2, n+3):
    if is_nondec(n, b): return b
print([a(n) for n in range(86)]) # Michael S. Branicky, Jun 01 2021

a(n) <= A000196(n) + 2.
a(n) <= 10 for any n in A009996.
a(n) = 2 iff n belongs to A023758.

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

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.

A357030 a(n) is the number of integers in 0..n having nonincreasing digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 19, 19, 19, 19, 19, 19, 19, 20, 21, 22, 23, 24, 24, 24, 24, 24, 24, 25, 26, 27, 28, 29, 30, 30, 30, 30, 30, 31, 32, 33, 34, 35, 36, 37, 37, 37, 37, 38, 39, 40

Offset: 0

Osman Mustafa Quddusi, Sep 09 2022

			a(20) = 13 because there are 13 numbers in 0..20 whose digits are in nonincreasing order: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20.

Cf. A009996.

a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+`if`(
     (l-> is(l=sort(l)))(convert(n, base, 10)), 1, 0))
    end:
seq(a(n), n=0..72);  # Alois P. Heinz, Sep 12 2022

Accumulate @ Table[If[GreaterEqual @@ IntegerDigits[n], 1, 0], {n, 0, 72}] (* Amiram Eldar, Sep 10 2022 *)

a(n) = sum(k=0, n, my(d=digits(k)); d == vecsort(d,,4)); \\ Michel Marcus, Sep 10 2022

cp's OEIS Frontend

A261020 Numbers k such that the set of the decimal digits is a subgroup of the multiplicative group (Z/mZ)* where m is the sum of the decimal digits of k.

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A261021 a(1)=0; for n > 1, a(n) is the number k such that the set of the decimal digits is an additive group Z/mZ where m is the sum of the decimal digits.

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A273046 Fibonacci numbers with digits in nonincreasing order.

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Mathematica

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A355063 Perfect powers whose digits are in nonincreasing order and do not include 0.

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A378808 Numbers with monotonically decreasing digits, decreasing by only 0 or 1.

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A265324 Numbers with nonincreasing digits such that every cyclic shift is a prime.

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A344887 a(n) is the least base k >= 2 that the base-k digits of n are nonincreasing.

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A364831 Primes whose digits are prime and in nonincreasing order.

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