A090055 -id:A090055 - cp's OEIS frontend

A090056 Numbers k divisible by at least one nontrivial permutation (rearrangement) of the digits of k, excluding all permutations that result in digit loss.

3105, 7128, 7425, 8316, 8712, 9513, 9801, 30105, 31050, 37125, 42741, 44172, 67128, 70416, 71208, 71253, 71280, 71328, 71928, 72108, 72441, 74142, 74250, 74628, 74925, 78912, 79128, 80712, 81816, 82755, 83160, 83181, 83916, 84510, 85725, 86712, 87120, 87132, 87192, 87912

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial permutations are identified as those where the permutation = k itself. Digit loss occurs when a permutation has 0 in the most significant position, which drops off, leaving a number with fewer digits. For example, when k is 3105, the permutation 0315 is excluded because 315 has fewer digits than 3105. These exclusions make this sequence a subsequence of A090055. A084687 is a subsequence of this sequence.
Apparently each term of this sequence is divisible by 3. This has been confirmed for the first 100 terms.
From David A. Corneth, Jun 08 2025: (Start)
All terms are divisible by 3. Proof: Suppose a term t is not divisible by 3.
Then for some m < t that is an anagram of t with the same number of digits as t we have m * c = t where 2 <= c <= 9. If c > 9 then t has more digits than m and if c = 1 then m is a trivial anagram of t, excluded by definition.
Since m and t are anagrams, 9 | (t - m) = (c - 1)*m. If t is not divisible by 3 then m is not divisible by 3 and so 9 | c - 1. This is a contradiction since 2 <= c <= 9 for which no c is divisible by 9 which completes the proof.
In addition if t is not divisible by 9 then c = 4 or 7. (End)

			a(1)=3105 because 3105 is divisible by 1035, a nontrivial permutation of 3105 with the same number of digits.
a(4)=8316 because 8316 is divisible by 1386, a nontrivial permutation of 8316 with the same number of digits.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 362 terms from Harvey P. Dale)
David A. Corneth, PARI program
C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A084687, A090055, A090061, A179239.

dnpQ[n_]:=Module[{d=FromDigits/@Select[Permutations[IntegerDigits[n]], First[#]>0&&Reverse[#]!=#&]},Count[Divisible[n,d],True]>1]; Select[ Range[90000],dnpQ] (* Harvey P. Dale, Aug 19 2013 *)

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

More terms from David A. Corneth, Jun 08 2025

A090057 Numbers n divisible by exactly two nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

1050, 1080, 3105, 5100, 5400, 7020, 7030, 9207, 9801, 10010, 10050, 10080, 10098, 10200, 10206, 20020, 20160, 20250, 20304, 20400, 20500, 20790, 21000, 21060, 30015, 30030, 30105, 30240, 30420, 30450, 30600, 35100, 40040, 40050

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.

			a(3)=3105 because 3105 is divisible by both 135 and 1035, two nontrivial permutations of 3105. a(8)=9207 because 9207 is divisible by both 279 and 297, two nontrivial permutations of 9207.

C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090058, A090059, A090060, A090061.

A090053 Numbers divisible by the number formed when their digits are sorted in ascending order, excluding trivial cases.

Original entry on oeis.org

105, 108, 405, 510, 540, 702, 703, 810, 1001, 1005, 1008, 1020, 1050, 1080, 2002, 2016, 2025, 2040, 2050, 2100, 3003, 3042, 3060, 3105, 3510, 4004, 4005, 4050, 4080, 4200, 5005, 5010, 5040, 5049, 5100, 5130, 5200, 5400, 6006, 6084

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial cases are identified as (1) values of k where the digits are already in ascending order, like 123 or 2228, such that ASort(k)=k, or (2) values of k where k mod 10 = 0 and all digits other than trailing zeros are in ascending order, like 12000 or 333500, such that ASort(k)=k/10^z, where z = the number of trailing zeros of k. In case (1), k/ASort(k) is equivalent to k/k (as in 123/123). In case (2), k/ASort(k) is 10^z (as in 12000/12). Neither of these cases is very interesting.

Sequence A084687 is a subsequence of this sequence, but that sequence excludes any value of k with 1 or more zero digits.

			a(1)=105 because the digits of 105 in ascending order are 015 and 105 is divisible by 15. 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, A090055, A090056.

A090058 Numbers n divisible by exactly three nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

4050, 8100, 10500, 10530, 10800, 34020, 51000, 54000, 60912, 68040, 70300, 80190, 95040, 100100, 100500, 100800, 102000, 105030

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.

			60912 is a term because 60912 is divisible by 1269, 1296 and 1692, three nontrivial permutations of 60912.

Chuck Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090057, A090059, A090060, A090061.

A090059 Numbers n divisible by exactly four nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

31050, 50490, 70200, 92070, 100035, 100980, 102060, 105000, 108000

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.

			50490 is a term because 50490 is divisible by 459, 495, 594 and 4590, four nontrivial permutations of 50490.

Chuck Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090057, A090058, A090060, A090061.

A090060 Numbers n divisible by exactly five nontrivial permutations (rearrangements) of the digits of n.

Original entry on oeis.org

40500, 81000, 98010, 105300

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.

			98010 is a term because 98010 is divisible by 198, 891, 1089, 8910 and 10890, five nontrivial permutations of 98010.

Chuck Seggelin, Numbers Divisible by Digit Permutations.

Cf. A090055, A090057, A090058, A090059, A090061.

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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A090056 Numbers k divisible by at least one nontrivial permutation (rearrangement) of the digits of k, excluding all permutations that result in digit loss.

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A090057 Numbers n divisible by exactly two nontrivial permutations (rearrangements) of the digits of n.

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A090053 Numbers divisible by the number formed when their digits are sorted in ascending order, excluding trivial cases.

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A090058 Numbers n divisible by exactly three nontrivial permutations (rearrangements) of the digits of n.

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A090059 Numbers n divisible by exactly four nontrivial permutations (rearrangements) of the digits of n.

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A090060 Numbers n divisible by exactly five nontrivial permutations (rearrangements) of the digits of n.

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

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