A235717 -id:A235717 - cp's OEIS frontend

A031955 Numbers with exactly two distinct base-10 digits.

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166

Offset: 1

Clark Kimberling

The three-digit terms are given by A210666(1,...,244). For numbers with exactly two distinct (but unspecified) digits in other bases, see A031948-A031954. For numbers made of two *given* digits, see A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases). - M. F. Hasler, Apr 04 2015
A235154 is a subsequence. - Altug Alkan, Dec 03 2015
A235717 is a subsequence. - Robert Israel, Dec 03 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Different from A029742.

Cf. A043638, A101594, A031948, A031949, A031950, A031951, A031952, A031953, A031954, A235154, A235717.

a031955 n = a031955_list !! (n-1)
a031955_list = filter ((== 2) . a043537) [0..]
-- Reinhard Zumkeller, Feb 05 2012

M:= 5: # to get all terms < 10^M
sort([seq(seq(seq(seq(add(10^(m-j)*`if`(member(j,S2),d2,d1),j=1..m)  ,
  S2 = combinat:-powerset({$2..m}) minus {{}}),
  d2 = {$0..9} minus {d1}), d1 = 1..9), m=2..M)]); # Robert Israel, Dec 03 2015

Select[Range@ 166, Length@ Union@ IntegerDigits@ # == 2 &] (* Michael De Vlieger, Dec 03 2015 *)

is_A031955(n)=#Set(digits(n))==2 \\ M. F. Hasler, Apr 04 2015

def ok(n): return len(set(str(n))) == 2
print(list(filter(ok, range(167)))) # Michael S. Branicky, Oct 12 2021

A043537(a(n)) = 2. - Reinhard Zumkeller, Dec 03 2009

Name edited by Charles R Greathouse IV, Feb 13 2017

A225218 Square numbers containing all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Original entry on oeis.org

1026753849, 1042385796, 1098524736, 1237069584, 1248703569, 1278563049, 1285437609, 1382054976, 1436789025, 1503267984, 1532487609, 1547320896, 1643897025, 1827049536, 1927385604, 1937408256, 2076351489, 2081549376, 2170348569, 2386517904, 2431870596

Offset: 1

Reiner Moewald, May 02 2013

The first term having a repeated digit is 10057482369. - Colin Barker, Jan 15 2014

			1026753849 is in the sequence because 1026753849 = 32043^2 and 1026753849 contains all ten digits 0, ..., 9.

Reiner Moewald, Table of n, a(n) for n = 1..4865

Supersequence of A036745.

Cf. A235717-A235724.

Select[#^2 &[Range[1000000]], Length[Union[IntegerDigits[#]]] == 10 &] (* Geoffrey Critzer, Jan 04 2015 *)

s=[]; for(n=1, 100000, if(#vecsort(eval(Vec(Str(n^2))), , 8)==10, s=concat(s, n^2))); s \\ Colin Barker, Jan 15 2014

from itertools import count, islice
def c(n): return len(set(str(n))) == 10
def agen(): yield from (k*k for k in count(31622) if c(k*k))
print(list(islice(agen(), 21))) # Michael S. Branicky, Dec 27 2022

a(n) = A054038(n)^2. - Colin Barker, Jan 15 2014

A235718 Squares which have one or more occurrences of exactly three different digits.

Original entry on oeis.org

169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, 961, 1156, 1225, 1521, 1600, 1681, 2025, 2116, 2209, 2500, 3136, 3364, 3600, 3844, 3969, 4225, 4489, 4624, 4900, 5625, 5776, 5929, 6400, 6561, 6889, 7225, 8100, 8281, 8464, 8836, 9409, 10201, 10404

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 1156.

			5929 is in the sequence because 5929 = 77^2 and 5929 contains exactly three different digits: 2, 5 and 9.

Giovanni Resta, Table of n, a(n) for n = 1..12299 (terms < 10^36, first 600 terms from Colin Barker)

Cf. A054031, A225218, A235717, A235719-A235724.

s=[]; for(n=1, 200, if(#vecsort(eval(Vec(Str(n^2))),,8)==3, s=concat(s, n^2))); s

a(n) = A054031(n)^2.

A235724 Squares which have one or more occurrences of exactly nine different digits.

Original entry on oeis.org

102495376, 102576384, 102738496, 104325796, 105637284, 139854276, 152843769, 157326849, 158306724, 158407396, 172843609, 176039824, 176305284, 178035649, 180472356, 183467025, 187635204, 198753604, 208571364, 215384976, 217356049, 218034756, 235714609

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 1005397264.

The smallest penholodigital square is a(6) = A036744(1) = 139854276 and the largest one is a(83) = A036744(30) = 923187456 (see Penguin references). - Bernard Schott, Feb 07 2022

			102495376 is in the sequence because 102495376 = 10124^2 and 102495376 contains exactly nine different digits: 0, 1, 2, 3, 4, 5, 6, 7 and 9.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 139854276, page 184 and entry 923187456, page 186.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Colin Barker)

Cf. A054037.
Cf. A235717-A235723, A225218.
A036744 is a subsequence.

s=[]; for(n=1, 100000, if(#vecsort(eval(Vec(Str(n^2))),,8)==9, s=concat(s, n^2))); s

from itertools import count, islice
def agen(): yield from (r*r for r in count(10**4) if len(set(str(r*r)))==9)
print(list(islice(agen(), 23))) # Michael S. Branicky, May 24 2022

a(n) = A054037(n)^2.

A335843 a(n) is the number of n-digit positive integers with exactly two distinct base 10 digits.

Original entry on oeis.org

0, 81, 243, 567, 1215, 2511, 5103, 10287, 20655, 41391, 82863, 165807, 331695, 663471, 1327023, 2654127, 5308335, 10616751, 21233583, 42467247, 84934575, 169869231, 339738543, 679477167, 1358954415, 2717908911, 5435817903, 10871635887, 21743271855, 43486543791

Offset: 1

Stefano Spezia, Jul 18 2020

a(n) is the number of n-digit numbers in A031955.

			a(1) = 0 since the positive integers must have at least two digits;
a(2) = 81 since #[99] - #[9] - #(11*[9]) = 99 - 9 - 9 = 81;
a(3) = 243 since #[999] - #[99] - #(111*[9]) - #{xyz in N | x,y,z are three different digits with x != 0} = 999 - 99 - 9 - 9*9*8 = 243;
...

Stefano Spezia, Table of n, a(n) for n = 1..3300
Puzzle Critic, Twitter post about the case n = 5, Jul 16 2020.
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index entries for sequences related to digits.

Cf. A000225, A031955, A337127, A337313.

Cf. A055642, A235154, A235690, A235717.

Mathematica
```
LinearRecurrence[{3,-2},{0,81},31]
```

concat([0],Vec(81*x^2/(1-3*x+2*x^2)+O(x^31)))

O.g.f.: 81*x^2/(1 - 3*x + 2*x^2).
E.g.f.: 81*(exp(x) - 1)^2/2.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
a(n) = 81*(2^(n-1) - 1).
a(n) = 81*A000225(n-1).

a(0) removed by Stefano Spezia, Sep 23 2020

A235723 Squares which have one or more occurrences of exactly eight different digits.

Original entry on oeis.org

10278436, 10673289, 10679824, 10837264, 13498276, 13527684, 13675204, 13860729, 13942756, 16378209, 16785409, 17280649, 17430625, 19847025, 20584369, 20738916, 21307456, 21473956, 21743569, 23078416, 23174596, 23970816, 24137569, 24671089, 24870169, 28901376

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 101465329.

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

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

Cf. A235717-A235722, A235724, A225218.

s=[]; for(n=1, 10000, if(#vecsort(eval(Vec(Str(n^2))),,8)==8, s=concat(s, n^2))); s

a(n) = A054036(n)^2.

A235719 Squares which have one or more occurrences of exactly four different digits.

Original entry on oeis.org

1024, 1089, 1296, 1369, 1764, 1849, 1936, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3249, 3481, 3721, 4096, 4356, 4761, 5041, 5184, 5329, 5476, 6084, 6241, 6724, 7056, 7396, 7569, 7921, 8649, 9025, 9216, 9604, 9801, 10609, 10816, 11025, 11236, 12544, 12996

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 10609.

			5329 is in the sequence because 5329 = 73^2 and 5329 contains exactly four different digits: 2, 3, 5 and 9.

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

Cf. A235717, A235718, A235720-A235724, A225218.

Select[Range[150]^2,Length[Union[IntegerDigits[#]]]==4&] (* Harvey P. Dale, May 03 2018 *)

s=[]; for(n=1, 300, if(#vecsort(eval(Vec(Str(n^2))),,8)==4, s=concat(s, n^2))); s

a(n) = A054032(n)^2.

A235720 Squares which have one or more occurrences of exactly five different digits.

Original entry on oeis.org

12769, 13456, 13689, 13924, 15376, 15876, 16384, 17689, 17956, 18496, 18769, 20164, 20736, 21609, 21904, 23104, 23409, 23716, 28561, 29584, 30276, 30625, 30976, 31684, 32041, 32761, 34596, 35721, 36481, 37249, 38025, 38416, 39204, 39601, 41209, 43681, 45369

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 100489.

			30976 is in the sequence because 30976 = 176^2 and 30976 contains exactly five different digits: 0, 3, 6, 7 and 9.

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

Cf. A235717-A235719, A235721-A235724, A225218.

s=[]; for(n=1, 1200, if(#vecsort(eval(Vec(Str(n^2))),,8)==5, s=concat(s, n^2))); s

a(n) = A054033(n)^2.

A235721 Squares which have one or more occurrences of exactly six different digits.

Original entry on oeis.org

103684, 104329, 104976, 107584, 123904, 124609, 132496, 134689, 139876, 140625, 157609, 162409, 164025, 170569, 173056, 180625, 195364, 198025, 207936, 209764, 214369, 237169, 254016, 257049, 258064, 259081, 279841, 293764, 310249, 318096, 321489, 326041

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 1028196.

			124609 is in the sequence because 124609 = 353^2 and 124609 contains exactly six different digits: 0, 1, 2, 4, 6 and 9.

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

Cf. A235717-A235720, A235722-A235724, A225218.

s=[]; for(n=1, 1200, if(#vecsort(eval(Vec(Str(n^2))),,8)==6, s=concat(s, n^2))); s

a(n) = A054034(n)^2.

A235722 Squares which have one or more occurrences of exactly seven different digits.

Original entry on oeis.org

1034289, 1046529, 1048576, 1054729, 1056784, 1073296, 1075369, 1085764, 1238769, 1247689, 1354896, 1380625, 1382976, 1432809, 1507984, 1605289, 1607824, 1630729, 1695204, 1708249, 1750329, 1763584, 1803649, 1827904, 1836025, 1890625, 1946025, 1974025

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 10137856.

			1247689 is in the sequence because 1247689 = 1117^2 and 1247689 contains exactly seven different digits: 1, 2, 4, 6, 7, 8 and 9.

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

Cf. A235717-A235721, A235723, A235724, A225218.

sddQ[n_]:=Count[DigitCount[n],?(#>0&)]==7; Select[Range[1001,1450]^2, sddQ] (* _Harvey P. Dale, Mar 12 2015 *)

s=[]; for(n=1, 10000, if(#vecsort(eval(Vec(Str(n^2))),,8)==7, s=concat(s, n^2))); s

a(n) = A054035(n)^2.

cp's OEIS Frontend

A031955 Numbers with exactly two distinct base-10 digits.

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A225218 Square numbers containing all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

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A235718 Squares which have one or more occurrences of exactly three different digits.

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A235724 Squares which have one or more occurrences of exactly nine different digits.

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A335843 a(n) is the number of n-digit positive integers with exactly two distinct base 10 digits.

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A235723 Squares which have one or more occurrences of exactly eight different digits.

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A235719 Squares which have one or more occurrences of exactly four different digits.

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