A009995 -id:A009995 - cp's OEIS frontend

A262557 Numbers with digits in strictly decreasing order, sorted lexicographically.

0, 1, 10, 2, 20, 21, 210, 3, 30, 31, 310, 32, 320, 321, 3210, 4, 40, 41, 410, 42, 420, 421, 4210, 43, 430, 431, 4310, 432, 4320, 4321, 43210, 5, 50, 51, 510, 52, 520, 521, 5210, 53, 530, 531, 5310, 532, 5320, 5321, 53210, 54, 540, 541, 5410, 542, 5420, 5421

Offset: 1

N. J. A. Sloane, Oct 14 2015

Original name: "Countdown sequences, allowing gaps."
Only digits 0 through 9 are used. The last term is 9876543210.
Equals A009995, sorted lexicographically. - Reinhard Zumkeller, Oct 14 2015
There are 2^k terms starting with digit k >= 0, they start at index 2^k. The countdown sequences, i.e., digits of the n-th term, are given in rows of A272011. - M. F. Hasler, Dec 11 2019

Donald S. McDonald, Email message to N. J. A. Sloane, Oct 14 2015.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1023

Cf. A009995, A263327, A263328.

a262557 n = a262557_list !! (n-1)
a262557_list = 0 : f [[0]] where
   f xss = if x < 9 then (map (read . concatMap show) zss) ++ f zss else []
           where zss = (map (z :) $ map tail xss) ++ (map (z :) xss)
                 z = x + 1; x = head $ head xss
-- Reinhard Zumkeller, Oct 14 2015

A262557[n_] := FromDigits[BitLength[n] - Flatten[Position[IntegerDigits[n, 2], 1]]]; Array[A262557, 100] (* or *)
A262557full = Rest[Map[FromDigits, LexicographicSort[Subsets[Range[9, 0, -1]]]]] (*  Paolo Xausa, Feb 13 2024 *)

is_A262557 = is_A009995
apply( A262557(n)=fromdigits(Vecrev(vecextract([0..exponent(n+!n)],n))), [1..99])
# A262557=concat(apply(x(i)=concat(vector(i%10+1,j,if(j>1,x(i*10+j-2),i))),[0..9])) \\ M. F. Hasler, Dec 11 2019

from itertools import combinations
afull = list(map(int, sorted("".join(c) for i in range(1, 11) for c in combinations("9876543210", i)))) # Michael S. Branicky, Feb 13 2024

a(n) = A009995(A263328(n)); a(A263327(n)) = A009995(n). - Reinhard Zumkeller, Oct 15 2015

New name from M. F. Hasler, Dec 11 2019

A190219 Numbers all of whose divisors have decimal digits in strictly decreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 31, 40, 41, 43, 53, 61, 62, 63, 71, 73, 82, 83, 86, 93, 97, 421, 431, 521, 541, 631, 641, 643, 653, 743, 751, 761, 821, 842, 853, 862, 863, 941, 953, 961, 971, 983, 5431, 6421, 6521, 7321, 7541, 7621, 7643, 8431, 8521

Offset: 1

Jaroslav Krizek, May 06 2011

Sequence is finite. Last term a(104) = 98765431.

Subset of A009995 and A190220. Superset of A052014.

			Number 93 is in sequence because all divisors of 93 (1, 3, 31, 93) are numbers whose decimal digits are in strictly decreasing order.

Nathaniel Johnston, Table of n, a(n) for n = 1..104 (full sequence)

with(numtheory): A190219 := 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(mA190219(n),n=1..60); # Nathaniel Johnston, May 06 2011

Select[Range[9000],Max[Flatten[Differences/@(IntegerDigits/@Divisors[#])]]<0&] (* Harvey P. Dale, Feb 22 2024 *)

A119252 Odd digits in decreasing order.

Original entry on oeis.org

1, 3, 5, 7, 9, 31, 51, 53, 71, 73, 75, 91, 93, 95, 97, 531, 731, 751, 753, 931, 951, 953, 971, 973, 975, 7531, 9531, 9731, 9751, 9753, 97531

Offset: 1

Zak Seidov, May 10 2006

There is a full list of 31 such numbers. Cf. A119253 Odd digits in increasing order, A009993 Digits in increasing order, A009995 Digits in decreasing order.

Cf. A009993, A009995, A119253.

Sort@Flatten@Table[FromDigits/@Subsets[Range[9,1,-2],{n}],{n,1,5}]

A119253 Odd digits in increasing order.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 35, 37, 39, 57, 59, 79, 135, 137, 139, 157, 159, 179, 357, 359, 379, 579, 1357, 1359, 1379, 1579, 3579, 13579

Offset: 1

Zak Seidov, May 10 2006

This is the complete list of all 31 such numbers.

Cf. A119252 Odd digits in decreasing order, A009993 Digits in increasing order, A009995 Digits in decreasing order.

Flatten@Table[FromDigits/@Subsets[Range[1,9,2],{n}],{n,1,5}]

A256825 Numbers with digits in strictly decreasing order containing digit 1.

Original entry on oeis.org

1, 10, 21, 31, 41, 51, 61, 71, 81, 91, 210, 310, 321, 410, 421, 431, 510, 521, 531, 541, 610, 621, 631, 641, 651, 710, 721, 731, 741, 751, 761, 810, 821, 831, 841, 851, 861, 871, 910, 921, 931, 941, 951, 961, 971, 981, 3210, 4210, 4310, 4321, 5210, 5310, 5321

Offset: 1

Jaroslav Krizek, Apr 10 2015

a(n) = possible values of A256824(m) in increasing order where A256824(m) = reverse concatenation of distinct digits of all divisors of m in base 10.
There are precisely 512 terms. Maximal term is 9876543210.
Subsequence of A009995 (numbers with digits in strictly decreasing order).
See A256826 - the smallest number k such that A256824(k) = a(n).

			21 is in sequence because there are numbers m such that A256824(m) = 21 (for m = 2, 22, 121, 211, 2111, ...).

Jaroslav Krizek, Table of n, a(n) for n = 1..512 (complete list)

Cf. A009995, A095050, A243534, A256824, A256826.

[Row n = 1 … 1023; Column A: A(n) = A009995(n); Column B: B(n) =  =IF(FIND("1";A(n);1)>0;A(n)); Arrangement of column B]

lista(nn) = for (n=1, nn, if ((d = digits(n)) && ((sd=vecsort(d,,8))==Vecrev(d)) && vecsearch(sd, 1), print1(n, ", "))); \\ Michel Marcus, Apr 11 2015

A256826 a(n) = the smallest number k such that A256824(k) = A256825(n).

Original entry on oeis.org

1, 101, 2, 3, 41, 5, 61, 7, 181, 19, 202, 103, 23, 401, 4, 43, 505, 25, 15, 451, 601, 122, 163, 461, 1661, 107, 127, 37, 47, 157, 67, 1801, 281, 83, 1481, 5581, 1861, 187, 109, 29, 9, 149, 59, 619, 79, 89, 2003, 404, 403, 123, 10, 503, 115, 4051, 12451, 453

Offset: 1

Jaroslav Krizek, Apr 13 2015

A256824(n) = reverse concatenation of distinct digits of all divisors of n in base 10, A256825(n) = possible values of A256824(m) in increasing order.

Finite sequence with 512 terms. Maximal term is a(185) = 88511.

			a(11) = 202 because 202 is the smallest number k such that reverse concatenation of distinct digits of all divisors of k (i.e. 1, 2, 101, 202) in base 10 = A256824(k) = A256824(202) = A256825(11) = 210.

Jaroslav Krizek, Table of n, a(n) for n = 1..512 (complete list)

Cf. A009995, A095050, A243534, A256824, A256825.

A256826:=func; [A256826(n): n in[A256825(n)]];

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.

Original entry on oeis.org

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

A358100 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly decreasing order.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 30, 40, 80, 60, 252, 120, 240, 540, 360, 630, 420, 960, 1440, 840, 1260, 2880, 3360, 4320, 2520, 6720, 5040, 8640, 10080, 15120, 50400, 20160, 40320, 30240, 171360, 90720, 383040, 60480, 120960, 181440, 362880, 544320, 937440, 786240, 2056320

Offset: 1

Bernard Schott, Nov 01 2022

This sequence is finite since A009995 is finite with 1022 nonzero terms, hence the last term is a(1022) = lcm of the 1022 positive terms of A009995.

			For n=7, the divisors of 30 are {1, 2, 3, 5, 6, 10, 15, 30} of which 7 have their decimal digits in strictly decreasing order (all except 15). No integer < 30 has 7 such divisors, so a(7) = 30.

Cf. A009995, A190219, A358099, A358101.

Similar: A087997 (palindromic), A355303 (undulating), A357172 (increasing order).

s[n_] := DivisorSum[n, 1 &, Greater @@ IntegerDigits[#] &]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = s[n]; If[i <= len && v[[i]] == 0, v[[i]] = n; c++]; n++]; v]; seq[45, 3*10^6] (* Amiram Eldar, Nov 01 2022 *)

f(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, , 12) == dd); \\ A358099
a(n) = my(k=1); while(f(k)!=n, k++); k; \\ Michel Marcus, Nov 01 2022

More terms from Amiram Eldar, Nov 01 2022

A358101 Positions of records in A358099, i.e., integers whose number of divisors whose decimal digits are in strictly decreasing order sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 30, 40, 60, 120, 240, 360, 420, 840, 1260, 2520, 5040, 8640, 10080, 15120, 20160, 30240, 60480, 120960, 181440, 362880, 544320, 786240, 1572480, 1874880, 3749760, 5624640, 7862400, 14938560, 23587200, 24373440, 31872960, 63745920, 95618880

Offset: 1

Bernard Schott, Nov 03 2022

As A009995 is finite, this sequence is necessarily finite.

Corresponding records are 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, ...

			a(9) = 60 is in the sequence because A358099(60) = 10 is larger than any earlier value in A358099.

Cf. A009995, A190219, A358099, A358100.

Similar sequences: A093036, A340548, A357173.

f[n_] := DivisorSum[n, 1 &, Greater @@ IntegerDigits[#] &]; fm = 0; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Nov 03 2022 *)

More terms from Amiram Eldar, Nov 03 2022

A377948 Numbers that have at least 1 repeated decimal digit and whose decimal digits are nondecreasing as place value decreases.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 333, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399

Offset: 1

Michael De Vlieger, Nov 14 2024

Intersection of A009994 and A109303.

Does not intersect either A009993 or A009995.

Michael De Vlieger, Table of n, a(n) for n = 1..10938 (includes all numbers up to 7 digits long)

Cf. A009993, A009994, A009995, A109303, A178788.

Select[Range[10^6], And[CountDistinct[#] != Length[#], AllTrue[Differences[#], # >= 0 &]] &[IntegerDigits[#]] &]
(* More efficient program: *)
b = 10; mm = b - 1; nn = 14;
s = Table[Map[Position[#, 1][[All, 1]] &,
  Permutations@ Join[ConstantArray[1, r], ConstantArray[0, mm - r] ] ],
    {r, Min[mm, nn]}];
Union@ Flatten@ Table[
  w = Apply[Join, Permutations /@ IntegerPartitions[n, Min[mm, n - 1] ] ];
  Reap[Do[
    Sow[Table[FromDigits[Flatten@
      MapIndexed[ConstantArray[m[[First[#2] ]], #1] &,
      w[[i]]], b], {m, s[[Length[w[[i]]] ]]} ] ],
    {i, Length[w]}] ][[-1, 1]], {n, 2, nn}]

A178788(a(n)) = 0.

cp's OEIS Frontend

A262557 Numbers with digits in strictly decreasing order, sorted lexicographically.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

A190219 Numbers all of whose divisors have decimal digits in strictly decreasing order.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

A119252 Odd digits in decreasing order.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A119253 Odd digits in increasing order.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A256825 Numbers with digits in strictly decreasing order containing digit 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Excel

PARI

A256826 a(n) = the smallest number k such that A256824(k) = A256825(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Sage

A358100 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly decreasing order.

Views

Author

Keywords

Comments

Examples