A055792 -id:A055792 - cp's OEIS frontend

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

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).

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).

A207359 Indices n, not squarefree, where A055231(n) = A055231(n-A055231(n)).

Original entry on oeis.org

9, 45, 63, 99, 117, 153, 171, 207, 261, 279, 289, 315, 333, 369, 387, 423, 477, 495, 531, 549, 585, 603, 639, 657, 676, 693, 711, 747, 765, 801, 819, 855, 873, 909, 927, 963, 981, 1017, 1035, 1071, 1143, 1179, 1197, 1233, 1251, 1287, 1305, 1341, 1359, 1395

Offset: 1

Michel Lagneau, Feb 17 2012

nonn

A055231(n) is the powerfree part of n. This sequence is infinite because all numbers of the form n = 9p, where p is a prime > 3, are in the sequence : A055231(9p) = p and A055231(9p - p) = A055231(8p) = p. The positive numbers of A055792 are also in the sequence because A055792(n) are squares and A055792(n)-1 are also squares.

			63 is in the sequence because A055231(63) = A055231(7*3^2) = 7, A055231(63 - 7) = A055231(56) = A055231(7*2^3) = 7.

Cf. A055231, A140394.

isA013929 := proc(n)
    n>3 and not numtheory[issqrfree](n) ;
end proc:
isA207359 := proc(n)
    isA013929(n) and (A055231(n)- A055231(n- A055231(n))=0);
end proc:
for n from 1 to 5000 do
      if isA207359(n) then
          printf(`%d, `,n);
      end if;
end do: #  adapted from A140394.

cp's OEIS Frontend

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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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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A207359 Indices n, not squarefree, where A055231(n) = A055231(n-A055231(n)).

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