A090053 -id:A090053 - cp's OEIS frontend

A090055 Numbers n divisible by at least one nontrivial permutation (rearrangement) of the digits of n.

105, 108, 405, 510, 540, 702, 703, 810, 1001, 1005, 1008, 1020, 1050, 1053, 1080, 2002, 2016, 2025, 2040, 2050, 2079, 2100, 2106, 3003, 3024, 3042, 3045, 3060, 3105, 3402, 3510, 4004, 4005, 4050, 4070, 4080, 4200, 5005, 5010, 5040

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial permutations are identified as (1) permutation = n, or (2) when n mod 10=0, permutations of n's digits which result in shifting only trailing zeros to the most significant side of n where they drop off, such that permutation = n/10^z, where z <= the number of trailing zeros of n. So if n were 1809000, the following permutations would be excluded as trivial: 1809000, 0180900, 0018090, 0001809.

A031877 (numbers which are multiples of their reversals) and both A084687 and A090053 (numbers divided by number formed by sorting their digits), are subsets of this sequence. This sequence differentiates itself by including terms such as 7425 which is divided by 2475 (a rearrangement of 7425's digits that is neither a reversal or an ascending sort.)

			a(27)=3045 because 3045 is divisible by 435, a nontrivial permutation of 3045. (0435)

C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A031877, A084687, A090056, A090057, A090058, A090059, A090060.

A090054 Numbers n which divide the number formed when their digits are sorted in descending order excluding trivial cases.

Original entry on oeis.org

1750842, 15922035, 17508420, 19750842

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial cases are identified as those values of n where the digits are already in descending order, like 3210 or 8222, such that DSort(n)=n. In such cases DSort(n)/n is equivalent to n/n (as in 3210/3210).

a(1) and a(2) are primitive. Clearly if DSort(n) mod n = 0, then dsort(n x 10) mod (n x 10) = 0. Therefore since 1750842 is a member, so will be 17508420, 175084200, 1750842000 and so on. The nonprimitive member 19750842 sets up the implication that 1(9...)750842 is a member. A quick test of 199750842, 1999750842 and 19999750842 seems to confirm this.

			a(1)=1750842 because the digits of 1750842 in descending order are 8754210 which is divisible by 1750842. a(24)=3105 because the digits of 3105 in ascending order are 135 and 3105 is divisible by 135.

C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A084687, A090053.

sdoQ[n_] := Module[{rs = FromDigits[ReverseSort[ IntegerDigits[n]]]},
rs != n && Divisible[rs, n]]; Select[Range[198*10^5],sdoQ] (* Harvey P. Dale, Sep 15 2021 *)

A342143 Take a(n), sort its digits into ascending order, divide the larger of the two numbers by the smaller and keep only the remainder: this remainder is present in a(n) as a substring of its digits.

Original entry on oeis.org

10, 20, 30, 40, 50, 52, 60, 70, 80, 90, 98, 100, 105, 106, 108, 110, 120, 130, 140, 150, 160, 170, 180, 186, 190, 198, 200, 205, 220, 230, 240, 250, 251, 260, 270, 274, 280, 290, 298, 300, 302, 330, 340, 350, 360, 370, 380, 390, 398, 400, 405, 410, 440, 450, 460, 470, 480, 490, 498, 500, 502, 510, 511

Offset: 1

Eric Angelini and Carole Dubois, Mar 01 2021

In sorting a number, leading zeros are erased.

This is the lexicographically earliest sequence of distinct positive terms with this property.

			a(1) = 10, which sorted is 1 (leading zeros are erased); 10/1 leaves a remainder 0, which is present in a(1);
a(2) = 20, which sorted is 2 (leading zeros are erased); 20/2 leaves a remainder 0, which is present in a(2);
...
a(6) = 52, which sorted is 25; 52/25 leaves a remainder 2, which is present in a(6); etc.

Cf. A090053.

lst={};k=1;Do[While[!StringContainsQ[ToString@k,ToString@Mod[#2,#]&@@(Sort@{k,FromDigits@Sort@IntegerDigits@k})],k++];AppendTo[lst,k];k++,{n,62}];lst (* Giorgos Kalogeropoulos, May 08 2022 *)

A386501 Numbers k divisible by A004719(k), excluding trivial cases.

Original entry on oeis.org

105, 108, 405, 1001, 1005, 1008, 1020, 2002, 2025, 2040, 2050, 3003, 3060, 4004, 4005, 4080, 5005, 6006, 6075, 7007, 7050, 8008, 9009, 10005, 10008, 10020, 10032, 10065, 10098, 10101, 10125, 10206, 10250, 16005, 19008, 20007, 20025, 20040, 20050

Offset: 1

Anuraag Pasula and Walter Robinson, Jul 23 2025

Trivial cases are identified as (1) values of k where there are already no 0s besides leading 0s, like 255 or 1296, such that A004719(k)=k, or (2) where k mod 10 = 0 and k/10 is already in the sequence or is itself a trivial case, like 10080 or 2550. In case (1), k/A004719(k) is equivalent to k/k (as in 255/255). In case (2), k/A004719(k) = 10 * (k/10)/A004719(k/10) when we already know that (k/10)/A004719(k/10) is already an integer (as in 1080/18).
Any number k of the form 1|(at least one 0)|5, such as 105 or 10000000005, will be included in this sequence because k will always be divisible by 3 and 5 due to divisibility rules, and thus will be divisible by A004719(k)=15.
Numbers of form 1|(at least one 0)|8, such as 108 or 10000008, or 4|(at least one 0)|5, such as 405 or 400005, will be included in this sequence for similar reasons.

			A004719(108)=18, 108/18=6.
A004719(9009)=99, 9009/99=91.
A004719(2040)=24, 2040/24=85, 2040 is nontrivial because 204/24=17/2.
50 is trivial because 50/10 = 5, and 5 is trivial because A004719(5)=5.

Subset of A090055.
Cf. A004719, A052382.
Cf. A096093, A090053.

def removeZeros(number):
    stringNum = str(number)
    stringNum = stringNum.replace("0", "")
    return int(stringNum)
for x in range(1, 100000):
    smallInt = removeZeros(x)
    if smallInt == x:
        continue
    if x % smallInt == 0:
        if x % 10 == 0:
            if (x//10) % removeZeros(x//10) == 0:
                continue
        print(x)

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A090055 Numbers n divisible by at least one nontrivial permutation (rearrangement) of the digits of n.

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A090054 Numbers n which divide the number formed when their digits are sorted in descending order excluding trivial cases.

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A342143 Take a(n), sort its digits into ascending order, divide the larger of the two numbers by the smaller and keep only the remainder: this remainder is present in a(n) as a substring of its digits.

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A386501 Numbers k divisible by A004719(k), excluding trivial cases.

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