A023110 -id:A023110 - cp's OEIS frontend

A204504 A204512(n)^2 = floor[A055872(n)/8]: Squares such that appending some digit in base 8 yields another square.

0, 0, 0, 1, 4, 36, 144, 1225, 4900, 41616, 166464, 1413721, 5654884, 48024900, 192099600, 1631432881, 6525731524, 55420693056, 221682772224, 1882672131025, 7530688524100, 63955431761796, 255821727047184, 2172602007770041, 8690408031080164, 73804512832419600

Offset: 1

M. F. Hasler, Jan 15 2012

Base-8 analog of A202303.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9),
A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7),
A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5),
A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3),
A001541=sqrt(A055792) (base 2).

b=8;for(n=1,2e9,issquare(n^2\b) & print1((n^2\b)","))

a(n)=polcoeff(x^4*(1 + 4*x + x^2 + 4*x^3)/(1 - 35*x^2 + 35*x^4 - x^6+O(x^n)), n)

a(n)=A204512(n)^2.

G.f. = x^4*(1 + 4*x + x^2 + 4*x^3)/(1 - 35*x^2 + 35*x^4 - x^6)

A204575 Squares such that [a(n)/2] is again a square (A055792), written in binary.

Original entry on oeis.org

0, 1, 1001, 100100001, 10011001001001, 1010001010010000001, 101011001001001011001001, 10110111001100110101000100001, 1100001001111011011000010110001001, 110011100111010101001010101001000000001

Offset: 1

M. F. Hasler, Jan 16 2012

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

A204576 Floor[A055792(n-1)/2]=A084703(n-2) (truncated squares), written in binary.

Original entry on oeis.org

0, 0, 100, 10010000, 1001100100100, 101000101001000000, 10101100100100101100100, 1011011100110011010100010000, 110000100111101101100001011000100, 11001110011101010100101010100100000000, 1101101100101100000000000111010111101000100

Offset: 1

M. F. Hasler, Jan 16 2012

A204575 with the last (binary) digit (necessarily = 1, except for a(1)=0) deleted.

Also: Squares, written in binary, such that appending a (binary) digit (necessarily = 1) yields another square (except for a(1)=0 which corresponds to A204575(1)=00, the only square which remains square when multiplied by 2).

David W. Wilson, Table of n, a(n) for n = 1..100

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

A204513 A204517(n)^2 = floor[A055859(n)/7]: Squares which written in base 7, with some digit appended, yield another square.

Original entry on oeis.org

0, 0, 0, 1, 9, 36, 289, 2304, 9216, 73441, 585225, 2340900, 18653761, 148644864, 594579456, 4737981889, 37755210249, 151020840996, 1203428746081, 9589674758400, 38358699033600, 305666163522721, 2435739633423369, 9742958533693476, 77638002106025089, 618668277214777344, 2474673108859109376, 19719746868766849921, 157139306672920022025, 628557226691680088100, 5008738066664673854881, 39912765226644470817024, 159651060906577883268096

Offset: 1

M. F. Hasler, Jan 15 2012

Base-7 analog of A202303.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

b=7;for(n=0,200,issquare(n^2\b) & print1(n^2\b,","))

A204513(n)=polcoeff((x^4 + 9*x^5 + 36*x^6 + 34*x^7 + 9*x^8 + 36*x^9 + x^10)/(1 - 255*x^3 + 255*x^6 - x^9+O(x^n)),n)

G.f. = (x^4 + 9*x^5 + 36*x^6 + 34*x^7 + 9*x^8 + 36*x^9 + x^10)/(1 - 255*x^3 + 255*x^6 - x^9)

A204577 Sqrt(floor[A204575(n)/2]), written in binary.

Original entry on oeis.org

0, 0, 10, 1100, 1000110, 110011000, 100101001010, 11011000100100, 10011101110001110, 1110010111100110000, 1010011101111110010010, 111101000000111000111100, 101100011100111010111010110

Offset: 1

M. F. Hasler, Jan 16 2012

D. W. Wilson, Table of n, a(n) for n = 1..100

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

A224955 Numbers that are not squares, but can become squares by prepending or appending one additional digit.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 12, 14, 19, 21, 22, 24, 28, 29, 32, 40, 41, 44, 48, 52, 56, 57, 61, 62, 67, 69, 72, 76, 78, 84, 89, 90, 96, 102, 108, 115, 116, 122, 129, 136, 152, 156, 160, 168, 176, 184, 193, 202, 209, 211, 216, 220, 230, 240, 241, 249, 250, 260, 270, 280

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 21 2013

There are potentially 15 ways for each number to become a square--by prepending a digit between 1 and 9, or appending one of {0,1,4,5,6,9}. However, only 74 of the first 10000 terms can become a square in more than one way.

			a(4)=6 because, though 6 is not a square, it can become a square by prepending a 1 to become 16. We can also obtain 36 and 64.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, List of squares that can be formed by concatenating one digit to the first 10000 terms.
Christian N. K. Anderson, Ulam spiral of a(n), with brighter colors corresponding to the number of ways a term may become a square.

Cf. A023110, A000290.

isA224955 := proc(n)
    local p,ndgs;
    if issqr(n) then
        return false;
    else
        ndgs := convert(n,base,10) ;
        for p from 1 to 9 do
            [op(ndgs),p] ;
            add(op(i,%)*10^(i-1),i=1..nops(%)) ;
            if issqr(%) then
                return true;
            end if;
        end do:
        for p in {0,1,4,5,6,9} do
            [p,op(ndgs)] ;
            add(op(i,%)*10^(i-1),i=1..nops(%)) ;
            if issqr(%) then
                return true;
            end if;
        end do:
        return false;
    end if;
end proc:
n := 1;
c := 1;
while n <= 10000 do
    if isA224955(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
    c := c+1 ;
end do: # R. J. Mathar, Mar 14 2016

Module[{nn=300,pre=Range[9],app={0,1,4,5,6,9}},Select[Range[nn],(!IntegerQ[ Sqrt[ #]]) && (AnyTrue[Sqrt[pre*10^IntegerLength[#]+#],IntegerQ] || AnyTrue[ Sqrt[ 10#+app],IntegerQ])&]] (* Harvey P. Dale, Feb 27 2022 *)

A203719 A204521(n)^2 = floor[A055812(n)/5]: Squares which written in base 5, with some digit appended, yield another square.

Original entry on oeis.org

0, 0, 0, 1, 9, 16, 64, 441, 3025, 5184, 20736, 142129, 974169, 1669264, 6677056, 45765225, 313679521, 537497856, 2149991424, 14736260449, 101003831721, 173072640400, 692290561600, 4745030099481, 32522920134769, 55728852710976, 222915410843904

Offset: 1

M. F. Hasler, Jan 16 2012

Base-5 analog of A202303.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

b=5;for(n=0,1e7,issquare(n^2\b) & print1(n^2\b,","))

Conjecture: a(n) = 323*a(n-4)-323*a(n-8)+a(n-12) for n>13. - Colin Barker, Sep 20 2014

Empirical g.f.: -x^4*(x^9 +9*x^8 +64*x^7 +16*x^6 +118*x^5 +118*x^4 +64*x^3 +16*x^2 +9*x +1) / ((x -1)*(x +1)*(x^2 -4*x -1)*(x^2 +1)*(x^2 +4*x -1)*(x^4 +18*x^2 +1)). - Colin Barker, Sep 20 2014

More terms from Colin Barker, Sep 20 2014

A204573 A204519(n)^2 = floor(A055851(n)/6): Squares which written in base 6, with some digit appended, yield another square.

Original entry on oeis.org

0, 0, 0, 1, 4, 16, 121, 400, 1600, 11881, 39204, 156816, 1164241, 3841600, 15366400, 114083761, 376437604, 1505750416, 11179044361, 36887043600, 147548174400, 1095432263641, 3614553835204, 14458215340816, 107341182792481, 354189388806400, 1416757555225600

Offset: 1

M. F. Hasler, Jan 16 2012

Base-6 analog of A202303.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

b=6;for(n=0,1e7,issquare(n^2\b) & print1(n^2\b,","))

Conjecture: a(n) = 99*a(n-3)-99*a(n-6)+a(n-9) for n>10. - Colin Barker, Sep 20 2014

Empirical g.f.: -x^4*(x^6+16*x^5+4*x^4+22*x^3+16*x^2+4*x+1) / ((x-1)*(x^2+x+1)*(x^6-98*x^3+1)). - Colin Barker, Sep 20 2014

A204574 Numbers such that floor[a(n)^2/2] is a square (A001541), written in binary.

Original entry on oeis.org

0, 1, 11, 10001, 1100011, 1001000001, 110100100011, 100110010010001, 11011111001000011, 10100010100100000001, 1110110011011111000011, 1010110010010010110010001, 111110110111010100110100011, 101101110011001101010001000001

Offset: 1

M. F. Hasler, Jan 16 2012

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

cp's OEIS Frontend

A204504 A204512(n)^2 = floor[A055872(n)/8]: Squares such that appending some digit in base 8 yields another square.

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A204575 Squares such that [a(n)/2] is again a square (A055792), written in binary.

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A204576 Floor[A055792(n-1)/2]=A084703(n-2) (truncated squares), written in binary.

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A204513 A204517(n)^2 = floor[A055859(n)/7]: Squares which written in base 7, with some digit appended, yield another square.

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A204577 Sqrt(floor[A204575(n)/2]), written in binary.

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A224955 Numbers that are not squares, but can become squares by prepending or appending one additional digit.

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Mathematica

A203719 A204521(n)^2 = floor[A055812(n)/5]: Squares which written in base 5, with some digit appended, yield another square.

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A204573 A204519(n)^2 = floor(A055851(n)/6): Squares which written in base 6, with some digit appended, yield another square.

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A204574 Numbers such that floor[a(n)^2/2] is a square (A001541), written in binary.

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