A016069 -id:A016069 - cp's OEIS frontend

A018885 Squares using no more than two distinct digits.

0, 1, 4, 9, 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

Offset: 1

David W. Wilson

Is 6661661161 the largest term not of the form 10^k, 4*10^k or 9*10^k? Any larger ones must have >= 22 digits. - Robert Israel, Dec 03 2015

Shawn A. Broyles, Table of n, a(n) for n = 1..85
Alexandru Gica and Laurentiu Panaitopol, On Oblath's Problem, J. Integer Seqs., Vol. 6(3), 2003, article 03.3.5.
Eric Weisstein's World of Mathematics, Square Number

Cf. A016069, A016070, A018884.

F:= proc(r, a, b, m)
# get all squares starting with r, with at most m further digits, all from {a,b} where a < b
local res,Ls,Us,L,U,looking;
if issqr(r) then res:= r else res:= NULL fi;
if m = 0 then return res fi;
Ls:= r*10^m + a*(10^m-1)/9;
Us:= r*10^m + b*(10^m-1)/9;
L:= isqrt(Ls);
if L^2 > Ls then L:= L-1 fi;
U:= isqrt(Us);
if U^2 < Us then U:= U+1 fi;
if L > U then res
else res, procname(10*r+a,a,b,m-1), procname(10*r+b,a,b,m-1)
fi
end proc:
S2:= {seq(i^2 mod 100, i=0..99)}:
prs:= map(t -> `if`(t < 10, {0,t},{(t mod 10),(t - (t mod 10))/10}), S2):
prs:= map(p -> `if`(nops(p)=1, seq(p union {s},s={$0..9} minus p), p), prs):
Res:= NULL:
for p in prs do
  a:= min(p); b:= max(p);
  if a > 0 then
     Res:= Res, F(a,a,b,14);
  fi;
  Res:= Res, F(b,a,b,14);
od:
sort(convert({0,Res},list)); # Robert Israel, Dec 03 2015

Select[Range[0, 10^5]^2, Length@ Union@ IntegerDigits@ # <= 2 &] (* Michael De Vlieger, Dec 03 2015 *)
Select[Range[0,100000]^2,Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jul 25 2020 *)

for (n=0, 10^6, if ( #Set(digits(n^2))<=2, print1(n^2, ", ") ) ); \\ Michel Marcus, May 21 2015

For n > 4, a(n) = A016069(n-4)^2.

0 inserted and definition edited by Jon E. Schoenfield, Jan 15 2014

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]

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 12, 16, 17, 33, 34, 48, 54, 285, 333, 334, 365, 385, 430, 471, 516, 816, 1049, 3333, 3334, 33333, 33334, 333333, 333334, 483048, 3333333, 3333334, 33333333, 33333334, 333333333, 333333334, 3333333333, 3333333334, 33333333333, 33333333334

Offset: 1

Seiichi Manyama, Aug 29 2019

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

Cf. A002277, A016069, A093137, A213517 (in case of triangular numbers), A270710, A322571.

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

Select[Range@ 50000, Length@ Union@ IntegerDigits[3 #^2 + 2 # - 1] <= 2 &] (* Giovanni Resta, Sep 04 2019 *)

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

For k > 0, A002277(k) is a term.

For k >= 0, A002277(k) + 1 (= A093137(k)) is a term.

a(35)-a(36) from Jinyuan Wang, Aug 30 2019

a(37)-a(40) from Giovanni Resta, Sep 04 2019

A380974 Numbers k such that k*(k-1) is composed of exactly two different decimal digits.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 17, 24, 25, 32, 34, 67, 75, 78, 100, 101, 142, 167, 334, 667, 1000, 1001, 1667, 3334, 6667, 10000, 10001, 16667, 33334, 66667, 100000, 100001, 166667, 333334, 666667, 1000000, 1000001, 1666667, 3333334, 6666667, 10000000, 10000001, 16666667, 33333334, 66666667, 100000000

Offset: 1

Robert Israel, Feb 11 2025

Numbers k such that A002378(k-1) is in A031955.
Conjecture: all terms >= 334 are of the form 10...0, 10...01, 16...67, 3...34, or 6...67.
Last decimal digit of a(n)*(a(n)-1) is either 0, 2 or 6. - Chai Wah Wu, Feb 19 2025

			a(10) = 23 is a term because 23 * 24 = 552 contains two different digits 2 and 5.

Michael S. Branicky, Table of n, a(n) for n = 1..117

Cf. A002378, A031955, A016069, A380984.

select(k -> nops(convert(convert(k*(k+1),base,10),set)) = 2, [$1..10^6]);

Select[Range[10^7],Length[Union[IntegerDigits[#*(#-1)]]]==2&] (* James C. McMahon, Feb 13 2025 *)

isok(k) = #Set(digits(k*(k-1))) == 2; \\ Michel Marcus, Feb 11 2025

from math import isqrt
from itertools import count, combinations, product, islice
def A380974_gen(): # generator of terms
    for n in count(1):
        c = []
        for a in combinations('0123456789',2):
            if '0' in a or '2' in a or '6' in a:
                for b in product(a,repeat=n):
                    if b[0] != '0' and b[-1] in {'0','2','6'} and b != (a[0],)*n and b != (a[1],)*n:
                        m = int(''.join(b))
                        q = isqrt(m)
                        if q*(q+1)==m:
                            c.append(q+1)
        yield from sorted(c)
A380974_list = list(islice(A380974_gen(),30)) # Chai Wah Wu, Feb 19 2025

Conjectured: for k >= 0,
  a(20 + 5*k) = (10^(3+k) + 2)/6,
  a(21 + 5*k) = (10^(3+k) + 2)/3,
  a(22 + 5*k) = (2*10^(3+k)+1)/3,
  a(23 + 5*k) = 10^(3+k),
  a(24 + 5*k) = 10^(3+k) + 1.
Conjectured G.f.: (4*x + 5*x^2 + 6*x^3 + 7*x^4 + 8*x^5 - 35*x^6 - 45*x^7 - 55*x^8 - 60*x^9 - 64*x^10 - 34*x^11 - 28*x^12 - 27*x^13 - 50*x^14 - 109*x^15 - 107*x^16 - 152*x^17 - 163*x^18 - 425*x^19 - 418*x^20 - 274*x^21 - 113*x^22 + 229*x^23 + 109*x^24 + 580*x^25 + 440*x^26 + 330*x^27 + 10*x^28 + 410*x^29)/(1 - 11 * x^5 + 10 * x^10).

A057659 Prime numbers whose square is composed of just two different decimal digits.

Original entry on oeis.org

5, 7, 11, 109, 173, 81619

Offset: 1

Carlos Rivera, Oct 15 2000

Conjectured to be a finite sequence.

No further terms up to prime(1000000)=15485863. - Harvey P. Dale, Mar 20 2011

			81619^2 = 6661661161.

Related to the Unsolved problem F24, p. 262, of the UPiNT by R. K. Guy

Subsequence of A016069.

Cf. A235154.

filter:= p -> nops(convert(convert(p^2,base,10),set))= 2:
select(filter, [seq(ithprime(i),i=1..10^5)]); # Robert Israel, Feb 11 2025

Select[Prime[Range[10000]],Count[DigitCount[#^2],0] ==8&] (* Harvey P. Dale, Mar 20 2011 *)

select(x->#Set(digits(x^2))==2, primes(10000)) \\ Michel Marcus, Feb 11 2025

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

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A018885 Squares using no more than two distinct 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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A322570 Positive integers k such that A270710(k) (= (k+1)*(3*k-1)) have only 1 or 2 different digits in base 10.

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A380974 Numbers k such that k*(k-1) is composed of exactly two different decimal digits.

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A057659 Prime numbers whose square is composed of just two different decimal digits.

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Maple

Mathematica

PARI

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

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.