A054038 -id:A054038 - cp's OEIS frontend

A054034 Numbers n such that n^2 contains exactly 6 different digits.

322, 323, 324, 328, 352, 353, 364, 367, 374, 375, 397, 403, 405, 413, 416, 425, 442, 445, 456, 458, 463, 487, 504, 507, 508, 509, 529, 542, 557, 564, 567, 571, 572, 574, 584, 589, 591, 593, 597, 598, 616, 618, 621, 625, 626, 629, 634, 637, 639, 645, 647

Offset: 1

Asher Auel, Feb 29 2000

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A016069, A054031, A054032, A054033, A054035, A054036, A054037, A054038, A054039.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=6 then f := [op(f),i] fi; od; f;

Select[Range[700],Count[DigitCount[#^2],0]==4&] (* Harvey P. Dale, May 10 2021 *)

A054035 Numbers n such that n^2 contains exactly 7 different digits.

Original entry on oeis.org

1017, 1023, 1024, 1027, 1028, 1036, 1037, 1042, 1113, 1117, 1164, 1175, 1176, 1197, 1228, 1267, 1268, 1277, 1302, 1307, 1323, 1328, 1343, 1352, 1355, 1375, 1395, 1405, 1428, 1433, 1441, 1442, 1444, 1463, 1541, 1593, 1594, 1628, 1646, 1648, 1701, 1706

Offset: 1

Asher Auel, Feb 29 2000

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A016069, A054031, A054032, A054033, A054034, A054036, A054037, A054038, A054039.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=7 then f := [op(f),i] fi; od; f;

Select[Range[2000],Count[DigitCount[#^2],0]==3&] (* Harvey P. Dale, Jul 28 2012 *)

A054036 Numbers n such that n^2 contains exactly 8 different digits.

Original entry on oeis.org

3206, 3267, 3268, 3292, 3674, 3678, 3698, 3723, 3734, 4047, 4097, 4157, 4175, 4455, 4537, 4554, 4616, 4634, 4663, 4804, 4814, 4896, 4913, 4967, 4987, 5376, 5529, 5699, 5742, 5853, 5899, 5904, 5905, 5968, 6043, 6071, 6095, 6098, 6127, 6176, 6181, 6199

Offset: 1

Asher Auel, Feb 28 2000

			3206 is in the sequence because 3206^2 = 10278436 and 10278436 contains exactly eight different digits: 0, 1, 2, 3, 4, 6, 7 and 8.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A016069, A054031, A054032, A054033, A054034, A054035, A054037, A054038, A054039, A235723.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=8 then f := [op(f),i] fi; od; f;

Select[Range[7000],Count[DigitCount[#^2],0]==2&] (* Harvey P. Dale, Aug 10 2017 *)

A258103 Number of pandigital squares (containing each digit exactly once) in base n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 4, 26, 87, 47, 87, 0, 547, 1303, 3402, 0, 24192, 187562

Offset: 2

Adam J.T. Partridge, May 20 2015

For n = 18, the smallest and largest pandigital squares are 2200667320658951859841 and 39207739576969100808801. For n = 19, they are 104753558229986901966129 and 1972312183619434816475625. For n = 20, they are 5272187100814113874556176 and 104566626183621314286288961. - Chai Wah Wu, May 20 2015
When n is even, (n-1) is a factor of the pandigital squares.  When n is odd, (n-1)/2 is a factor with the remaining factors being odd.  Therefore, when n is odd and (n-1)/2 has an odd number of 2s as prime factors there are no pandigital squares in base n (e.g. 5, 13, 17 and 21). - Adam J.T. Partridge, May 21 2015
If n is odd and (n-1)/2 has an odd 2-adic valuation, then there are no squares in base n using all the digits from 1 to n-1 once, or all the digits from 0 to n-2 once or all the digits from 1 to n-2 once. This can be proved using the same argument as in the linked blogposts. - Chai Wah Wu, Feb 25 2024

			For n=4 there is one pandigital square, 3201_4 = 225 = 15^2.
For n=6 there is one pandigital square, 452013_6 = 38025 = 195^2.
For n=10 there are 87 pandigital squares (A036745).
There are no pandigital squares in bases 2, 3, 5 or 13.
Hexadecimal has 3402 pandigital squares, the largest is FED5B39A42706C81.

A. J. T. Partridge, Why there are no pandigital squares in base 13
Chai Wah Wu, Square pandigital numbers
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 2.

Cf. A036745, A054038, A071519.

a(n) = if(n%2==1 && valuation(n-1,2)%2==0, 0, my(lim=sqrtint(n^n - (n^n-n)/(n-1)^2), count=0); for(m=sqrtint((n^n-n)/(n-1)^2 + n^(n-2)*(n-1) - 1), lim, if(#Set(digits(m^2,n))==n, count++)); count) \\ Jianing Song, Feb 23 2024. Note that the searching range for m is [sqrt(A049363(n)), sqrt(A062813(n))]

from gmpy2 import isqrt, mpz, digits
def A258103(n): # requires 2 <= n <= 62
    c, sm, sq = 0, mpz(''.join([digits(i, n) for i in range(n-1, -1, -1)]), n), mpz(''.join(['1', '0']+[digits(i, n) for i in range(2, n)]), n)
    m = isqrt(sq)
    sq = m*m
    m = 2*m+1
    while sq <= sm:
        if len(set(digits(sq, n))) == n:
            c += 1
        sq += m
        m += 2
    return c # Chai Wah Wu, May 20 2015

a(17)-a(19) from Giovanni Resta, May 20 2015

A363905 Numbers whose square and cube taken together contain each decimal digit.

Original entry on oeis.org

69, 128, 203, 302, 327, 366, 398, 467, 542, 591, 593, 598, 633, 643, 669, 690, 747, 759, 903, 923, 943, 1016, 1018, 1027, 1028, 1043, 1086, 1112, 1182, 1194, 1199, 1233, 1278, 1280, 1282, 1328, 1336, 1364, 1396, 1419, 1459, 1463, 1467, 1472, 1475

Offset: 1

M. F. Hasler, Jun 27 2023

The first term, a(1) = 69, is the only number for which the square and the cube together contain each decimal digit 0 to 9 exactly once.

a(820) = 6534 is the only number of which the square and cube taken together contain each digit 0 to 9 exactly twice.

			69^2 = 4761, 69^3 = 328509, which together contain each digit 0-9 exactly once.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Harold Suarez, Interesting..., Number Theory group on LinkedIn, June 2023.

Cf. A036744, A054038, A071519 and A156977 for "pandigital" squares.

Cf. A119735: Numbers n such that every digit occurs at least once in n^3.

fQ[n_] := Union[ Join[ IntegerDigits[n^2], IntegerDigits[n^3]]] == {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}; Select[Range@1500, fQ] (* Robert G. Wilson v, Jun 27 2023 *)

is(k)=#setunion(Set(digits(k^2)),Set(digits(k^3)))>9
select(is,[1..9999])

from itertools import count, islice
def A363905_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:len(set(str(n**2))|set(str(n**3)))==10,count(max(startvalue,1)))
A363905_list = list(islice(A363905_gen(),20)) # Chai Wah Wu, Jun 27 2023

A121321 Numbers k such that every digit occurs at least once in k^4.

Original entry on oeis.org

763, 767, 1066, 1088, 1206, 1304, 1425, 1557, 1561, 1634, 1653, 1712, 1739, 1782, 1818, 1839, 1866, 1878, 1975, 2032, 2045, 2055, 2134, 2192, 2232, 2233, 2299, 2318, 2339, 2347, 2358, 2425, 2441, 2489, 2509, 2511, 2575, 2646, 2682, 2692

Offset: 1

Tanya Khovanova, Aug 25 2006

			763^4 = 338920744561 contains every digits at least once.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A119735 (in k^3), A054038 (in k^2).

[ n: n in [0..2695] | Seqset(Intseq(n^4)) eq {0..9} ]; // Bruno Berselli, May 17 2011

Select[Range[2692],ContainsAll[IntegerDigits[#^4],Range[0,9]]&] (* James C. McMahon, Oct 16 2024 *)

PARI
```
isok(k) = #Set(digits(k^4)) == 10;
```

A154871 Numbers n such that n^6 contains every digit exactly 4 times.

Original entry on oeis.org

3470187, 3554463, 3887058, 4328241, 4497738

Offset: 1

Zhining Yang, Jan 16 2009

a(1) is also the number of A074205

			3887058^6=3449261536602702380309782557611971988544, which contains 4 times of each digit 0-9. Total 5 terms

A054038 A074205 A154532 A154566 A154818

A154873 Numbers n such that n^5 contains every digit exactly 3 times.

Original entry on oeis.org

643905, 680061, 720558, 775113, 840501, 878613, 984927

Offset: 1

Zhining Yang, Jan 16 2009

a(5) is also the number of A074205

			840501^5=419460598737334268928156702501, which contains exactly 3 times of each digit 0-9. Total 7 terms

A054038 A074205 A154532 A154566 A154818

Select[Range[630972,999978],Union[DigitCount[#^5]]=={3}&] (* Harvey P. Dale, May 01 2021 *)

A154874 Numbers k such that k^3 contains every digit exactly twice.

Original entry on oeis.org

2158479, 2190762, 2205528, 2219322, 2301615, 2330397, 2336268, 2345811, 2358828, 2359026, 2367609, 2388534, 2389119, 2389638, 2397132, 2428986, 2504736, 2524974, 2536152, 2583258, 2590125, 2607222, 2620827, 2622012, 2647866, 2649369, 2658636, 2671593

Offset: 1

Zhining Yang, Jan 16 2009

This sequence has 138 terms.

			2358828^3 = 13124683009764879552, which contains each digit 0..9 exactly twice.

Nathaniel Johnston, Table of n, a(n) for n = 1..138 (full sequence)

Cf. A054038, A074205, A154532, A154566, A154871, A154873.

lim:=floor((10^20)^(1/3)): for j from ceil((10^19)^(1/3)) to lim do d:=convert(j^3,base,10): doubdig:=true: for k from 0 to 9 do if(numboccur(d,k)<>2)then doubdig:=false:break: fi: od: if(doubdig)then print(j); fi: od: # Nathaniel Johnston, May 28 2011

With[{cmin=Ceiling[Surd[10^19,3]],cmax=Floor[Surd[10^20,3]]},Select[ Range[ cmin, cmax], Union[ DigitCount[#^3]]=={2}&]] (* Harvey P. Dale, Nov 17 2018 *)

A217368 Smallest number having a power that in decimal has exactly n copies of all ten digits.

Original entry on oeis.org

32043, 69636, 643905, 421359, 320127, 3976581, 47745831, 15763347, 31064268, 44626422, 248967789, 85810806, 458764971, 500282265, 2068553967, 711974055, 2652652791, 901992825, 175536645, 3048377607, 3322858521, 1427472867, 3730866429, 9793730157

Offset: 1

James G. Merickel, Oct 01 2012

The exponents that produce the number with a fixed number of copies of each digit are listed in sequence A217378. See there for further comments.
Since we allow A217378(n)=1, the sequence is well defined, with the upper bound a(n) <= 100...99 ~ 10^(10n-1) (n copies of each digit, sorted in increasing order, except for one "1" permuted to the first position). - M. F. Hasler, Oct 05 2012
What is the minimum value of a(n)? Can it be proved that a(n) > 2 for all n? - Charles R Greathouse IV, Oct 16 2012

			The third term raised to the fifth power (A217378(3)=5), 643905^5 = 110690152879433875483274690625, has three copies of each digit (in its decimal representation), and no number smaller than 643905 has a power with this feature.

Cf. A020666, A054038, A054039, A066825, A067635, A074205, A156977, A217378.

f[n_] := Block[{k = 2, t = Table[n, {10}], r = Range[0, 9]}, While[c = Count[ IntegerDigits[k^Floor[ Log[k, 10^(10 n)]]], #] & /@ r; c != t, k++]; k] (* Robert G. Wilson v, Nov 28 2012 *)

is(n,k)=my(v);for(e=ceil((10*n-1)*log(10)/log(k)), 10*n*log(10)/log(k), v=vecsort(digits(k^e)); for(i=1,9,if(v[i*n]!=i-1 || v[i*n+1]!=i, return(0))); return(1)); 0
a(n)=my(k=2); while(!is(n,k),k++); k \\ Charles R Greathouse IV, Oct 16 2012

a(13)-a(14) from James G. Merickel, Oct 06 2012 and Oct 08 2012
a(15)-a(16) from Charles R Greathouse IV, Oct 17 2012
a(17)-a(19) from Charles R Greathouse IV, Oct 18 2012
a(20) from Charles R Greathouse IV, Oct 22 2012
a(21)-a(24) from Giovanni Resta, May 05 2017

cp's OEIS Frontend

A054034 Numbers n such that n^2 contains exactly 6 different digits.

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A054035 Numbers n such that n^2 contains exactly 7 different digits.

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A054036 Numbers n such that n^2 contains exactly 8 different digits.

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Maple

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A258103 Number of pandigital squares (containing each digit exactly once) in base n.

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A363905 Numbers whose square and cube taken together contain each decimal digit.

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Mathematica

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A121321 Numbers k such that every digit occurs at least once in k^4.

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Magma

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A154871 Numbers n such that n^6 contains every digit exactly 4 times.

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A154873 Numbers n such that n^5 contains every digit exactly 3 times.

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Mathematica