A155146 -id:A155146 - cp's OEIS frontend

A155147 Numbers k such that k^3 has exactly 4 different digits.

12, 13, 16, 17, 18, 19, 21, 23, 25, 26, 28, 31, 33, 34, 37, 39, 42, 43, 44, 45, 46, 50, 53, 56, 58, 60, 64, 67, 68, 74, 77, 78, 79, 80, 81, 83, 86, 90, 91, 95, 98, 99, 106, 111, 114, 117, 121, 122, 132, 140, 142, 146, 150, 157, 160, 163, 164, 171, 172, 175, 177, 179

Offset: 1

Dmitry Kamenetsky, Jan 21 2009

John Cerkan, Table of n, a(n) for n = 1..1006

Cf. A155146.

a := proc (n) if nops(convert(convert(n^3, base, 10), set)) = 4 then n else end if end proc: seq(a(n), n = 1 .. 220); # Emeric Deutsch, Jan 26 2009

Extended by Emeric Deutsch, Jan 26 2009

A235807 Numbers n such that n^3 has one or more occurrences of exactly five different digits.

Original entry on oeis.org

22, 24, 27, 29, 32, 35, 38, 41, 47, 48, 49, 51, 52, 54, 55, 57, 61, 63, 65, 71, 72, 82, 85, 87, 89, 94, 96, 102, 103, 104, 105, 108, 109, 119, 120, 123, 125, 126, 127, 130, 133, 134, 136, 137, 138, 141, 143, 144, 149, 152, 153, 154, 155, 158, 162, 165, 167

Offset: 1

Colin Barker, Jan 19 2014

			22 is in the sequence because 22^3 = 10648, which contains exactly five different digits: 0, 1, 4, 6, 8.
87 is in the sequence because 87^3 = 658503, which contains exactly five different digits: 0, 3, 5, 6, 8.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A030292, A155146, A155147, A235808-A235811, A119735.

[n: n in [0..200] | #Set(Intseq(n^3)) eq 5]; // Bruno Berselli, Jan 19 2014

Select[Range[200], Length[Union[IntegerDigits[#^3]]] == 5 &] (* Bruno Berselli, Jan 19 2014 *)

s=[]; for(n=1, 200, if(#vecsort(eval(Vec(Str(n^3))),,8)==5, s=concat(s, n))); s

A235807_list, m = [], [6, -6, 1, 0]
for n in range(1,10**5+1):
    for i in range(3):
        m[i+1] += m[i]
    if len(set(str(m[-1]))) == 5:
        A235807_list.append(n) # Chai Wah Wu, Nov 05 2014

A235811 Numbers n such that n^3 has one or more occurrences of exactly nine different digits.

Original entry on oeis.org

1018, 1028, 1112, 1452, 1475, 1484, 1531, 1706, 1721, 1733, 1818, 1844, 1895, 1903, 2008, 2033, 2208, 2214, 2217, 2223, 2257, 2274, 2277, 2327, 2329, 2336, 2354, 2394, 2403, 2524, 2525, 2589, 2647, 2686, 2691, 2694, 2727, 2733, 2784, 2842, 2866, 2884, 2890

Offset: 1

Colin Barker, Jan 19 2014

			1018 is in the sequence because 1018^3 = 1054977832, which contains exactly nine different digits.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A030292, A155146, A155147, A235807-A235810, A119735.

Select[Range[3000],Count[DigitCount[#^3],0]==1&] (* Harvey P. Dale, Dec 17 2021 *)

s=[]; for(n=1, 3000, if(#vecsort(eval(Vec(Str(n^3))),,8)==9, s=concat(s, n))); s

A235811_list, m = [], [6, -6, 1, 0]
for n in range(1,10**4+1):
    for i in range(3):
        m[i+1] += m[i]
    if len(set(str(m[-1]))) == 9:
        A235811_list.append(n) # Chai Wah Wu, Nov 05 2014

A235808 Numbers k such that k^3 has one or more occurrences of exactly six different digits.

Original entry on oeis.org

59, 66, 69, 73, 75, 76, 84, 88, 93, 97, 107, 112, 113, 115, 116, 118, 124, 128, 129, 131, 139, 147, 148, 151, 156, 159, 161, 166, 168, 169, 174, 178, 181, 183, 184, 187, 189, 193, 194, 196, 207, 219, 226, 232, 234, 235, 236, 238, 240, 241, 246, 253, 255, 262

Offset: 1

Colin Barker, Jan 19 2014

			59 is in the sequence because 59^3 = 205379, which contains exactly six different digits: 0, 2, 3, 5, 7, 9.
107 is in the sequence because 107^3 = 1225043, which contains exactly six different digits: 0, 1, 2, 3, 4, 5.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A030292, A155146, A155147, A235807, A235809-A235811, A119735.

[n: n in [0..300] | #Set(Intseq(n^3)) eq 6]; // Bruno Berselli, Jan 19 2014

Select[Range[300], Length[Union[IntegerDigits[#^3]]] == 6 &] (* Bruno Berselli, Jan 19 2014 *)
Select[Range[300],Count[DigitCount[#^3],0]==4&] (* Harvey P. Dale, May 28 2025 *)

s=[]; for(n=1, 300, if(#vecsort(eval(Vec(Str(n^3))),,8)==6, s=concat(s, n))); s

A235808_list, m = [], [6, -6, 1, 0]
for n in range(1,10**4+1):
    for i in range(3):
        m[i+1] += m[i]
    if len(set(str(m[-1]))) == 6:
        A235808_list.append(n) # Chai Wah Wu, Nov 05 2014

A235809 Numbers k such that k^3 has one or more occurrences of exactly seven different digits.

Original entry on oeis.org

135, 145, 203, 221, 223, 225, 227, 233, 243, 244, 245, 247, 249, 254, 257, 265, 272, 273, 275, 276, 299, 313, 327, 329, 334, 338, 341, 345, 347, 352, 365, 366, 368, 382, 384, 388, 393, 395, 398, 403, 405, 409, 411, 412, 434, 439, 447, 452, 455, 473, 486, 493

Offset: 1

Colin Barker, Jan 19 2014

			135 is in the sequence because 135^3 = 2460375, which contains exactly seven different digits.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A030292, A155146, A155147, A235807, A235808, A235810, A235811, A119735.

[n: n in [0..1200] | #Set(Intseq(n^3)) eq 7]; // Vincenzo Librandi, Nov 07 2014

Select[Range[500], Length[Union[IntegerDigits[#^3]]]==7&] (* Vincenzo Librandi, Nov 07 2014 *)

s=[]; for(n=1, 600, if(#vecsort(eval(Vec(Str(n^3))),,8)==7, s=concat(s, n))); s

for(n=0,10^3,if(#Set(digits(n^3))==7,print1(n,", "))); \\ Joerg Arndt, Nov 10 2014

from itertools import count, islice
def A235809gen(): return filter(lambda n:len(set(str(n**3))) == 7,count(0))
A235809_list = list(islice(A235809gen(),26)) # Chai Wah Wu, Dec 23 2021

A235810 Numbers n such that n^3 has one or more occurrences of exactly eight different digits.

Original entry on oeis.org

289, 297, 302, 319, 467, 494, 515, 557, 562, 595, 621, 623, 676, 682, 709, 712, 721, 862, 887, 909, 939, 945, 949, 963, 984, 987, 1012, 1015, 1016, 1025, 1029, 1043, 1049, 1065, 1075, 1087, 1104, 1106, 1107, 1114, 1118, 1132, 1137, 1154, 1161, 1167, 1178

Offset: 1

Colin Barker, Jan 19 2014

			289 is in the sequence because 289^3 = 24137569, which contains exactly eight different digits.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A030292, A155146, A155147, A235807-A235809, A235811, A119735.

[n: n in [0..1200] | #Set(Intseq(n^3)) eq 8]; // Vincenzo Librandi, Nov 07 2014

s=[]; for(n=1, 1500, if(#vecsort(eval(Vec(Str(n^3))),,8)==8, s=concat(s, n))); s

A235810_list, m = [], [6, -6, 1, 0]
for n in range(1,10**3+1):
    for i in range(3):
        m[i+1] += m[i]
    if len(set(str(m[-1]))) == 8:
        A235810_list.append(n) # Chai Wah Wu, Nov 05 2014

A247045 Triangle read by rows: T(n,k) = least number m > 0 such that m^k in base n contains exactly k distinct digits, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 2, 3, 5, 1, 3, 3, 6, 12, 1, 3, 5, 7, 7, 15, 1, 3, 5, 9, 5, 17, 15, 1, 4, 5, 10, 9, 7, 11, 33, 1, 3, 5, 7, 11, 19, 14, 16, 53, 1, 4, 5, 6, 7, 13, 13, 14, 21, 36, 1, 4, 5, 7, 10, 8, 12, 12, 16, 42, 41, 1, 4, 6, 16, 11, 8, 19, 19, 16, 28, 35, 55, 1, 4, 6, 9, 9, 14, 10, 18, 14

Offset: 1

Derek Orr, Sep 10 2014

			T(n,k) is given by (row n corresponds to base n):
1;
1, 2;
1, 3, 4;
1, 2, 3,  5;
1, 3, 3,  6, 12;
1, 3, 5,  7,  7, 15;
1, 3, 5,  9,  5, 17, 15;
1, 4, 5, 10,  9,  7, 11, 33;
1, 3, 5,  7, 11, 19, 14, 16, 53;
1, 4, 5,  6,  7, 13, 13, 14, 21, 36; (base 10)
1, 4, 5,  7, 10,  8, 12, 12, 16, 42, 41;
Example: T(7,3) = 5 means that 5 is the smallest number such that 5^3 in base 7 (which is 125 in base 7 = 236) has 3 distinct digits (2, 3, and 6).

Indranil Ghosh, Rows 1..36 of triangle, flattened

Cf. A016069, A155146, A155150.

print1(1,", ");n=2;while(n<20,m=1;for(k=1,n,while(m,d=digits(m^k,n);if(#vecsort(d,,8)!=k,m++);if(#vecsort(d,,8)==k,print1(m,", ");m=1;break)));n++)

A247047 Numbers k such that k^2 contains exactly 2 distinct digits and k^3 contains exactly 3 distinct digits.

Original entry on oeis.org

5, 6, 8, 9, 15, 30, 173, 300, 3000, 30000, 300000, 3000000, 30000000, 300000000, 3000000000, 30000000000, 300000000000, 3000000000000, 30000000000000, 300000000000000, 3000000000000000

Offset: 1

Derek Orr, Sep 10 2014

Intersection of A016069 and A155146.
This sequence is infinite since 3*10^k is always in this sequence for k > 0.
Is 173 the last term not of the form 3*10^k?
3*10^7 < a(14) <= 3*10^8.
The numbers k such that k^2 contains 2 distinct digits, k^3 contains 3 distinct digits, and k^4 contains 4 distinct digits are conjectured to only be 6, 8, and 15. (Intersection of A016069, A155146, and A155150.)

			k = 15 is a member of this sequence since 15^2 = 225 contains two distinct digits and 15^3 = 3375 contains three distinct digits.

Cf. A016069, A155146.

Select[Range[3*10^6],Length[DeleteCases[DigitCount[#^2],0]]==2&&Length[ DeleteCases[ DigitCount[#^3],0]]==3&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Jan 21 2023 *)

for(n=1,3*10^7,d2=digits(n^2);d3=digits(n^3);if(#vecsort(d2,,8)==2&&#vecsort(d3,,8)==3,print1(n,", ")))

A247047_list = [n for n in range(1,10**6) if len(set(str(n**3))) == 3 and len(set(str(n**2))) == 2]
# Chai Wah Wu, Sep 26 2014

a(14)-a(15) from Chai Wah Wu, Sep 26 2014
a(16)-a(18) from Kevin P. Thompson, Jul 01 2022
a(19)-a(21) from Michael S. Branicky, Jun 05 2025

A385175 Cubes using at most three distinct digits, not ending in 0.

Original entry on oeis.org

1, 8, 27, 64, 125, 216, 343, 512, 729, 1331, 2744, 3375, 46656, 238328, 778688, 1030301, 5177717, 7077888, 9393931, 700227072, 1003003001, 44474744007, 1000300030001, 1000030000300001, 1331399339931331, 3163316636166336, 1000003000003000001, 1000000300000030000001, 1000000030000000300000001

Offset: 1

Gonzalo Martínez, Jun 20 2025

This sequence has infinitely many terms since (10^m + 1)^3 is a term for all m >= 0.

Conjecture: a(26) = 3163316636166336 is the largest term with nonzero digits (See comments of A030294 and the data of A155146, where a(26) = A155146(47)^3).

			8, 343, and 46656 belong to this list because they are cubes that use 1, 2, and 3 distinct digits, respectively.

Cf. A000578, A030292, A155146, A018884, A202940.

Select[Range[10^6]^3,Length[Union[IntegerDigits[#]]]<4&&IntegerDigits[#][[-1]]!=0&] (* James C. McMahon, Jun 30 2025 *)
fQ[n_] := Mod[n, 10] > 0 && Length@ Union@ IntegerDigits[n^3] < 4; k = 1; lst = {}; While[k < 1000002, If[ fQ@k, AppendTo[lst, k]]; k++]; lst^3 (* Robert G. Wilson v, Jul 10 2025 *)

a(n) = A202940(n)^3.

a(28) from Robert G. Wilson v, Jul 10 2025

a(29) from David A. Corneth, Jul 10 2025

cp's OEIS Frontend

A155147 Numbers k such that k^3 has exactly 4 different digits.

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A235807 Numbers n such that n^3 has one or more occurrences of exactly five different digits.

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A235811 Numbers n such that n^3 has one or more occurrences of exactly nine different digits.

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A235808 Numbers k such that k^3 has one or more occurrences of exactly six different digits.

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Magma

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A235809 Numbers k such that k^3 has one or more occurrences of exactly seven different digits.

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Magma

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A235810 Numbers n such that n^3 has one or more occurrences of exactly eight different digits.

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A247045 Triangle read by rows: T(n,k) = least number m > 0 such that m^k in base n contains exactly k distinct digits, 1 <= k <= n.

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A247047 Numbers k such that k^2 contains exactly 2 distinct digits and k^3 contains exactly 3 distinct digits.

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