A018885 -id:A018885 - cp's OEIS frontend

A016069 Numbers k such that k^2 contains exactly 2 distinct digits.

4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 21, 22, 26, 30, 38, 88, 100, 109, 173, 200, 212, 235, 264, 300, 1000, 2000, 3000, 3114, 10000, 20000, 30000, 81619, 100000, 200000, 300000, 1000000, 2000000, 3000000, 10000000, 20000000, 30000000, 100000000, 200000000, 300000000

Offset: 1

Robert G. Wilson v

10^k, 2*10^k, 3*10^k for k > 0 are terms. - Chai Wah Wu, Dec 17 2021

Subsequence of primes is A057659. - Bernard Schott, Jul 29 2022

			26 is in the sequence because 26^2 = 676 contains exactly 2 distinct digits.

R. K. Guy, Unsolved Problems in Number Theory, F24.

Robert G. Wilson v, Table of n, a(n) for n = 1..81
Eric Weisstein's World of Mathematics, Square Number

Cf. A016070, A018884, A018885, A057659.

import Data.List (nub)
a016069 n = a016069_list !! (n-1)
a016069_list = filter ((== 2) . length . nub . show . (^ 2)) [0..]
-- Reinhard Zumkeller, Apr 14 2011

[n: n in [0..20000000] | #Set(Intseq(n^2)) eq 2]; // Vincenzo Librandi, Nov 04 2014

Join[Select[Range[90000],Count[DigitCount[#^2],?(#!=0&)]==2&],Flatten[ NestList[ 10#&,{100000,200000,300000},5]]] (* _Harvey P. Dale, Mar 09 2013 *)
Select[Range[20000000], Length[Union[IntegerDigits[#^2]]]==2&] (* Vincenzo Librandi, Nov 04 2014 *)

/* needs version >= 2.6 */
for (n=1, 10^9, if ( #Set(digits(n^2))==2, print1(n,", ") ) );
/* Joerg Arndt, Mar 09 2013 */

from gmpy2 import is_square, isqrt
from itertools import combinations, product
A016069_list = []
for g in range(2,10):
    n = 2**g-1
    for x in combinations('0123456789',2):
        for i,y in enumerate(product(x,repeat=g)):
            if i > 0 and i < n and y[0] != '0':
                z = int(''.join(y))
                if is_square(z):
                    A016069_list.append(int(isqrt(z)))
A016069_list = sorted(A016069_list) # Chai Wah Wu, Nov 03 2014

a(n) = ((n-1) mod 3 + 1)*10^(ceiling(n/3)-7) for n >= 34 (conjectured). - Chai Wah Wu, Dec 17 2021

A235717 Squares which have one or more occurrences of exactly two different digits.

Original entry on oeis.org

16, 25, 36, 49, 64, 81, 100, 121, 144, 225, 400, 441, 484, 676, 900, 1444, 7744, 10000, 11881, 29929, 40000, 44944, 55225, 69696, 90000, 1000000, 4000000, 9000000, 9696996, 100000000, 400000000, 900000000, 6661661161, 10000000000, 40000000000, 90000000000

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 100.

This sequence is the same as A018885, except that A018885 has four additional leading terms.

			69696 is in the sequence because 69696 = 264^2 and 69696 contains exactly two different digits: 6 and 9.

Cf. A235718-A235724, A225218.

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

a(n) = A016069(n)^2.

A018884 Squares using at most two distinct digits, not ending in 0.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 225, 441, 484, 676, 1444, 7744, 11881, 29929, 44944, 55225, 69696, 9696996, 6661661161

Offset: 1

David W. Wilson

No other terms below 10^41.
The sequence is probably finite.
The two distinct digits of a term cannot both be in the set {0,2,3,7,8}. Looking at the digits (with leading zeros) of i^2 mod 10^4 for 0 <= i < 10^4 shows that there are no repunit terms > 10 and the two distinct digits of a term must be one of the following 21 pairs: '01', '04', '09', '12', '14', '16', '18', '24', '25', '29', '34', '36', '45', '46', '47', '48', '49', '56', '67', '69', '89'. - Chai Wah Wu, Apr 06 2019

Richard K. Guy, Unsolved Problems in Number Theory, Section F24 (at p. 262) (Springer-Verlag, 2d ed. 1994).

Eric Weisstein's World of Mathematics, Square Number.

Cf. A000290, A016069, A016070, A018885.

Flatten[Table[Select[Flatten[Table[FromDigits/@Tuples[{a,b},n],{n,10}]], IntegerQ[ Sqrt[#]]&],{a,9},{b,9}]]//Union (* Harvey P. Dale, Sep 21 2018 *)

A016070 Numbers k such that k^2 contains exactly 2 different digits, excluding 10^m, 210^m, 310^m.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 11, 12, 15, 21, 22, 26, 38, 88, 109, 173, 212, 235, 264, 3114, 81619

Offset: 1

Robert G. Wilson v

No other terms below 3.16*10^20 (derived from A018884).

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 109, p. 38, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, F24.

Michael Geißer, Theresa Körner, Sascha Kurz, and Anne Zahn, Squares with three digits, arXiv:2112.00444 [math.NT], 2021.
Eric Weisstein's World of Mathematics, Square Number.

Cf. A016069, A043537, A018884, A018885.

Select[Range[100000],Length[DeleteCases[DigitCount[#^2],0]]==2 && !Divisible[ #,10]&] (* Harvey P. Dale, Aug 15 2013 *)
Reap[For[n = 4, n < 10^5, n++, id = IntegerDigits[n^2]; If[FreeQ[id, {, 0 ...}], If[Length[Union[id]] == 2, Sow[n]]]]][[2, 1]] (* _Jean-François Alcover, Sep 30 2016 *)

from gmpy2 import is_square, isqrt
from itertools import combinations, product
A016070_list = []
for g in range(2,20):
    n = 2**g-1
    for x in combinations('0123456789',2):
        if not x in [('0','1'), ('0','4'), ('0','9')]:
            for i,y in enumerate(product(x,repeat=g)):
                if i > 0 and i < n and y[0] != '0':
                    z = int(''.join(y))
                    if is_square(z):
                        A016070_list.append(isqrt(z))
A016070_list = sorted(A016070_list) # Chai Wah Wu, Nov 03 2014

A043537(a(n)) = 2. [Reinhard Zumkeller, Aug 05 2010]

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.

Original entry on oeis.org

4, 15, 32, 55, 84, 119, 455, 799, 900, 3332, 3535, 7007, 8855, 244244, 333332, 335335, 400404, 445444, 555559, 666464, 799799, 1999199, 3303300, 33333332, 33353335, 3333333332, 3333533335, 333333333332, 333335333335, 700007077007, 33333333333332, 33333353333335, 3333333333333332, 3333333533333335

Offset: 1

Seiichi Manyama, Aug 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002277, A018885 (in case of squares), A213516 (in case of triangular numbers), A270710, A322570, A323639.

[a:k in [1..10000000]| #Set(Intseq(a)) le 2 where a is (k+1)*(3*k-1)]; // Marius A. Burtea, Aug 29 2019

Select[Table[(n+1)(3n-1),{n,3334*10^4}],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jun 12 2022 *)

for(k=1, 1e8, if(#Set(digits(j=3*k^2+2*k-1))<=2, print1(j", ")))

a(n) = A270710(A322570(n)).

For k > 0, A002277(2*k) - 1 is a term.

A368049 Perfect squares whose decimal expansion consists of k > 1 digits, k-1 of which are equal.

Original entry on oeis.org

16, 25, 36, 49, 64, 81, 100, 121, 144, 225, 400, 441, 484, 676, 900, 1444, 10000, 40000, 44944, 90000, 1000000, 4000000, 9000000, 100000000, 400000000, 900000000, 10000000000, 40000000000, 90000000000, 1000000000000, 4000000000000, 9000000000000, 100000000000000

Offset: 1

José Hernández, Dec 09 2023

The terms > 90000 are of one of the following three forms: 10^(2*j), 4*10^(2*j) or 9*10^(2*j) where j is an integer >= 3.

See Gica/Panaitopol link for proof of above comment. - Ray Chandler, Jan 16 2024

Ray Chandler, Table of n, a(n) for n = 1..1000
Alexandru Gica and Laurențiu Panaitopol, On Oblath's Problem, J. Integer Seq., Vol. 6 (2003), article 03.3.5, 12 pp.
Index entries for linear recurrences with constant coefficients, signature (0,0,100).

Cf. A235717, A018885.

Subsequence of A000290.

a(n) = 100*a(n-3) for n > 22. - Stefano Spezia, Dec 09 2023

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A016069 Numbers k such that k^2 contains exactly 2 distinct digits.

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A235717 Squares which have one or more occurrences of exactly two different digits.

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A018884 Squares using at most two distinct digits, not ending in 0.

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A016070 Numbers k such that k^2 contains exactly 2 different digits, excluding 10^m, 2*10^m, 3*10^m.

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A322571 Positive integers A270710(k) (= (k+1)*(3*k-1)) which have only 1 or 2 different digits in base 10.

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Programs

Magma

Mathematica

PARI

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A368049 Perfect squares whose decimal expansion consists of k > 1 digits, k-1 of which are equal.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A016070 Numbers k such that k^2 contains exactly 2 different digits, excluding 10^m, 210^m, 310^m.

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.