A136810 -id:A136810 - cp's OEIS frontend

A256630 Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 4 as largest digit.

142201, 1422010, 11141110, 11411110, 11412021, 14220100, 20323421, 21024111, 101203421, 110141011, 110142201, 111411100, 114111100, 114120210, 120013421, 141433102, 142201000, 203234210, 210241110, 1012034210, 1101410011, 1101410110, 1101422010, 1114111000

Offset: 1

Felix Fröhlich, Apr 05 2015

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A256631, A256633, A256634, A256708, A256709, A256889.

Subsequence of A136810.

fQ[n_] := Block[{c = DigitCount@ n}, And[Plus @@ Take[c, {5, 9}] == 0, c[[4]] > 0, c[[10]] > 0]]; Select[Range@ 10000000, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 12 2015 *)

is(n) = vecmin(digits(n))==0 && vecmin(digits(n^2))==0 && vecmax(digits(n))==4 && vecmax(digits(n^2))==4

from itertools import product
A256630_list = []
for l in range(11):
    for a in ('1','2','3','4'):
        for b in product('01234',repeat = l):
            for c in ('0','1','2'):
                s = a+''.join(b)+c
                if '0' in s and '4' in s:
                    n = int(s)
                    s2 = set(str(n**2))
                    if {'0','4'} <= s2 <= {'0','1','2','3','4'}:
                        A256630_list.append(n)
print(A256630_list) # Chai Wah Wu, Apr 17 2015

More terms from Alois P. Heinz, Apr 16 2015

A136809 Numbers k such that k and k^2 use only the digits 0, 1, 2 and 3.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11100, 11101, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111, 101000, 101001, 101010, 101011, 101100, 101110, 110000, 110001, 110010, 110100, 110101, 111000, 111001, 111010

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with "DrScheme".
Subsequence of A136810-A136815. - M. F. Hasler, Jan 24 2008
From M. F. Hasler, Nov 03 2020: (Start)
A136813(144) = 31733311 is the first term of that sequence which is not in this sequence. All others among A136810-A136815 differ much earlier.
If a(n) is a term then so is 10*a(n). Conjecture: the sequence contains only "binary numbers" (digits 0 or 1) having no more than three 1's in a row, and not more than one run of two or more consecutive 1's. (But not all such numbers, since 101101 for example is not in the sequence.) (End)
Here is a counterexample with two instances of two consecutive ones, i.e., two non-overlapping occurrences of the substring 11: 110010000010100001010000010011^2 = 12102200102222202222322212223022221222322220222220100220121. - Michael S. Branicky, Nov 04 2020
One cannot test candidates digit-by-digit from the right. Specifically, a suffix of the valid counterexample above is invalid: 10100001010000010011^2 = 102010020402001222322220222220100220121. - Michael S. Branicky, Nov 05 2020
With respect to the conjectures and comment above, only digits 0 and 1 occur and no 1111's occur in the first 83990 terms (all with <= 25 digits). These were generated incrementally from the right based on partial screening (see Python program). - Michael S. Branicky, Jul 07 2022

			111^2 = 12321,
11101^2 = 123232201, and
101011^2 = 10203222121,
111001^2 = 12321222001, so 111, 11101, 101011 and 111001 are in the sequence, but:
110011^2 = 12102420121, so 110011 is not in the sequence; also
1100011^2 = 1210024200121, so 1100011 is not in the sequence, and
1010101^2 = 1020304030201, so 1010101 is not in the sequence; but
1110001^2 = 1232102220001, so 1110001 is in the sequence; also
1010100100001^2 = 1020302212022030200200001.

Michael S. Branicky, Table of n, a(n) for n = 1..12670 (all terms with <= 19 digits; terms 1..1380 from Jonathan Wellons)
Jonathan Wellons, Tables of Shared Digits

Cf. A007088 (binary numbers), A136808 (subsequence: only 0, 1, 2), A136810, A136811, A136812, A136813, A136814, A136815.

Select[Range[0,200000],And@@(ContainsAll[{0,1,2,3},Union@IntegerDigits@#]&/@{#,#^2})&] (* Giorgos Kalogeropoulos, May 21 2021 *)
With[{c={0,1,2,3}},Select[FromDigits/@Tuples[c,6],SubsetQ[c,IntegerDigits[ #^2]]&]] (* Harvey P. Dale, Jun 01 2021 *)

select( {is_A136809(n,o(n)=vecmax(digits(n))<4)=o(n^2)&&o(n)}, [fromdigits(binary(n))|n<-[0..99]]) \\ M. F. Hasler, Nov 03 2020

from itertools import count, islice
def agen(only="0123"):
    digset, valid = set(only), set(only)
    for e in count(1):
        found, newvalid = set(), set()
        for tstr in valid:
            t = int(tstr)
            if (tstr == "0" or tstr[0] != "0") and set(str(t**2)) <= digset:
                found.add(t)
            for d in digset:
                dtstr = d + tstr
                dt = int(dtstr)
                remstr = str(dt**2)[-e:]
                if set(remstr) <= digset:
                    newvalid.add(dtstr)
        valid = newvalid
        yield from sorted(found)
print(list(islice(agen(), 50))) # Michael S. Branicky, Jul 07 2022

A136808 Numbers k such that k and k^2 use only the digits 0, 1 and 2.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10110, 11000, 11001, 11010, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101100, 110000, 110001, 110010, 110100, 1000000, 1000001, 1000010, 1000011, 1000100

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.
Subsequence of A136809, A136816, ..., A136836. - M. F. Hasler, Jan 24 2008
A278038(18) = 10101, A136827(294) = 10110001101, A136831(1276) = 101100010001101 resp. A136836(1262) = 101090009991101 are the first terms from where on these four sequences differ from the present one. - M. F. Hasler, Nov 15 2017
From Jovan Radenkovicc, Nov 15 2024: (Start)
A nonnegative integer n is in this sequence iff 10*n is also in this sequence.
Not a subsequence of A278038 (binary numbers without '111'). A counterexample is 10^2884 + 10^2880 + 10^2872 + 10^2857 + 10^2497 + 10^2426 + 10^2285 + 10^2004 + 10^1443 + 10^1442 + 10^1441 + 10^881 + 10^600 + 10^459 + 10^388 + 10^27 + 10^12 + 10^4 + 1. There are infinitely many counterexamples not divisible by 10. This counterexample follows from the fact that 111^2+2000*4+200*4=12321+8000+800=21121. In fact, every binary substring will eventually occur in this sequence. Also, if n is a term containing only the digits 0 and 1, then 10^k*n+1 and n+10^k are also in this sequence for any sufficiently large integer k. (End)

			101000100100001^2 = 10201020220210222010200200001.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1359 terms from Jonathan Wellons)
Jonathan Wellons, Tables of Shared Digits [archived]
Index to sequences related to squares.

A subsequence of the binary numbers A007088.
Cf. A278038.
Cf. A136809, A136810, ..., A137147 for other digit combinations.
See also A058412 = A058411^2: squares having only digits {0,1,2}, A277946 = A277959^2 = squares whose largest digit is 2.

isA136808 := proc(n) local ndgs,n2dgs ; ndgs := convert(convert(n,base,10),set) ; n2dgs := convert(convert(n^2,base,10),set) ; if ( (ndgs union n2dgs) minus {0,1,2} ) = {} then true ; else false ; fi ; end: LtonRev := proc(L) local i ; add(op(i,L)*10^(i-1),i=1..nops(L)) ; end: A007089 := proc(n) convert(n,base,3) ; LtonRev(%) ; end: n := 1: for i from 0 do n3 := A007089(i) ; if isA136808(n3) then printf("%d %d ",n,n3) ; n := n+1 ; fi ; od: # R. J. Mathar, Jan 24 2008

Select[FromDigits/@Tuples[{0,1},7],Union[Take[DigitCount[#^2],{3,9}]]=={0}&] (* Harvey P. Dale, May 29 2013 *)

for(n=1,999,vecmax(digits((N=fromdigits(binary(n),10))^2))<3 && print1(N",")) \\ M. F. Hasler, Nov 15 2017

A058412 Squares composed of digits {0,1,2}, not ending with zero.

Original entry on oeis.org

1, 121, 10201, 22201, 1002001, 1022121, 1212201, 100020001, 100220121, 121022001, 210221001, 10000200001, 10002200121, 10020210201, 10201202001, 12100220001, 100021020121, 1000002000001, 1000022000121, 1000202010201

Offset: 1

Patrick De Geest, Nov 15 2000

All terms but the first one have their largest digit equal to 2, cf. A277946 = A277959^2. - M. F. Hasler, Nov 15 2017

P. De Geest, Index to related sequences
Hisanori Mishima, Sporadic tridigital solutions
OEIS Index to sequences related to squares.

Cf. A058411.
Cf. A063009, A066139. - Zak Seidov, Jul 01 2013
Cf. A136808, A136809 and A136810, ..., A137147 for other digit combinations.
See also A277946 = A277959^2 = squares whose largest digit is 2.
The first 1261 terms are also a subsequence of A278038 (binary numbers without '111'), in turn a subsequence of the binary numbers A007088.

apply( t->t^2, vector(100,i,N=if(i>1,next_A058411(N),1))) \\ M. F. Hasler, Nov 15 2017

a(n) = A058411(n)^2. - Zak Seidov, Jul 01 2013

A158304 Numbers k such that k^2 contains no digit greater than 4.

Original entry on oeis.org

0, 1, 2, 10, 11, 12, 18, 20, 21, 32, 38, 48, 49, 100, 101, 102, 110, 111, 120, 149, 152, 179, 180, 182, 200, 201, 210, 318, 320, 321, 332, 338, 348, 351, 361, 362, 380, 451, 452, 462, 480, 482, 490, 548, 549, 649, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021

Offset: 1

N. J. A. Sloane, May 23 2010

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Erich Friedman, What's Special About This Number?

Cf. A158082, A136810.

A158304 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do d := convert(convert(a^2,base,10),set) ; if max(op(d)) <= 4 then return a; end if; end do; end if; end proc:
seq(A158304(n),n=1..100) ; # R. J. Mathar, May 23 2010

Select[Range[0,1100],Max[IntegerDigits[#^2]]<5&] (* Harvey P. Dale, Nov 06 2016 *)

Extended by several correspondents, May 24 2010

A136836 Numbers k such that k and k^2 use only the digits 0, 1, 2 and 9.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10110, 11000, 11001, 11010, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101100, 110000, 110001, 110010, 110100, 1000000, 1000001, 1000010, 1000011, 1000100

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.
Comparison of b-files indicates that the first difference from A136831 is at the 1262nd entry. - R. J. Mathar Apr 29 2008
More precisely, A278038(18) = 10101, A136827(294) = 10110001101 resp. A136808(1262) = A136831(1262) = 101100000000000 are the first terms from where on these four sequences differ from the present one; a(1262) = 101090009991101 is also the first term containing a digit > 1. - M. F. Hasler, Nov 15 2017

			101090009991101^2 = 10219190120000900002099192201.

Jonathan Wellons, Table of n, a(n) for n = 1..1360
J. Wellons, Tables of Shared Digits [archived]

Cf. A136808, A136809, A136810, ..., A137147 for other digit combinations.
See also A058412 = A058411^2: squares having only digits {0,1,2}, A277946 = A277959^2 = squares whose largest digit is 2.
The first 1261 terms are also a subsequence of A278038 (binary numbers without '111'), in turn a subsequence of the binary numbers A007088.

With[{c={0,1,2,9}},Select[FromDigits/@Tuples[c,7],SubsetQ[c,IntegerDigits[#^2]]&]] (* Harvey P. Dale, Feb 11 2024 *)

A257085 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..6.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 10, 11, 12, 15, 16, 20, 21, 25, 32, 34, 35, 40, 45, 46, 50, 51, 55, 56, 60, 65, 66, 100, 101, 102, 105, 106, 110, 111, 112, 115, 116, 120, 121, 125, 142, 145, 146, 150, 152, 155, 156, 160, 162, 200, 201, 204, 205, 206, 210, 211, 215, 216, 225, 231, 235, 245, 246, 250, 251, 252, 254, 255, 256

Offset: 1

Danny Rorabaugh, Apr 15 2015

			116 is in the list because 116 and 116^2 = 13456 do not use the digits 7, 8 or 9.
182 is not in the list because it uses the digit 8 (even though 182^2 = 33124 would be fine).
253 is not in the list because 253^2 = 6409 uses the digit 9.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A178501 (0..1), A136808(0..2), A136809(0..3), A136810 (0..4), A257086 (0..5).

Cf. A136859, A256633.

Select[Range@ 256, Total@ Take[DigitCount[#], {7, 9}] == 0 && Total@ Take[DigitCount[#^2], {7, 9}] == 0 &] (* Michael De Vlieger, Apr 17 2015 *)

isok(n) = ((vecmax(digits(n)) <= 6) && (vecmax(digits(n^2)) <= 6)) || (n==0); \\ Michel Marcus, Feb 02 2016

A257086 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..5.

Original entry on oeis.org

0, 1, 2, 5, 10, 11, 12, 15, 20, 21, 32, 35, 45, 50, 55, 100, 101, 102, 105, 110, 111, 112, 115, 120, 145, 150, 152, 155, 200, 201, 205, 210, 211, 235, 320, 321, 332, 335, 350, 351, 450, 451, 452, 500, 501, 502, 505, 550, 1000, 1001, 1002, 1005, 1010, 1011, 1012, 1015, 1020, 1021, 1050, 1055, 1100

Offset: 1

Danny Rorabaugh, Apr 15 2015

			115 is in the list because 115 and 115^2 = 13225 do not use the digits 6, 7, 8, or 9.
121 is not in the list because 121^2 = 14641 uses the digit 6.
149 is not in the list because it uses the digit 9 (even though 149^2 = 22201 would be okay).

Danny Rorabaugh, Table of n, a(n) for n = 1..5000

Cf. A178501 (0..1), A136808(0..2), A136809(0..3), A136810 (0..4), A257085 (0..6).

Cf. A136817, A256631.

Select[Range@ 1100, Total@ Take[DigitCount[#], {6, 9}] == 0 && Total@ Take[DigitCount[#^2], {6, 9}] == 0 &] (* Michael De Vlieger, Apr 17 2015 *)

cp's OEIS Frontend

A256630 Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 4 as largest digit.

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A136809 Numbers k such that k and k^2 use only the digits 0, 1, 2 and 3.

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A136808 Numbers k such that k and k^2 use only the digits 0, 1 and 2.

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A058412 Squares composed of digits {0,1,2}, not ending with zero.

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A158304 Numbers k such that k^2 contains no digit greater than 4.

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A136836 Numbers k such that k and k^2 use only the digits 0, 1, 2 and 9.

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A257085 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..6.

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A257086 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..5.

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